id sid tid token lemma pos ejde-39 1 1 electronic electronic ADJ ejde-39 1 2 journal journal NOUN ejde-39 1 3 of of ADP ejde-39 1 4 differential differential ADJ ejde-39 1 5 equations equation NOUN ejde-39 1 6 , , PUNCT ejde-39 1 7 vol vol NOUN ejde-39 1 8 . . PUNCT ejde-39 1 9 2023 2023 NUM ejde-39 1 10 ( ( PUNCT ejde-39 1 11 2023 2023 NUM ejde-39 1 12 ) ) PUNCT ejde-39 1 13 , , PUNCT ejde-39 1 14 no no INTJ ejde-39 1 15 . . NOUN ejde-39 1 16 23 23 NUM ejde-39 1 17 , , PUNCT ejde-39 1 18 pp pp PROPN ejde-39 1 19 . . PUNCT ejde-39 2 1 1–11 1–11 PROPN ejde-39 2 2 . . PUNCT ejde-39 3 1 issn issn PROPN ejde-39 3 2 : : PUNCT ejde-39 3 3 1072 1072 NUM ejde-39 3 4 - - SYM ejde-39 3 5 6691 6691 NUM ejde-39 3 6 . . PUNCT ejde-39 4 1 url url PROPN ejde-39 4 2 : : PUNCT ejde-39 4 3 https://ejde.math.txstate.edu https://ejde.math.txstate.edu PROPN ejde-39 4 4 or or CCONJ ejde-39 4 5 https://ejde.math.unt.edu https://ejde.math.unt.edu PROPN ejde-39 4 6 prescribed prescribe VERB ejde-39 4 7 energy energy NOUN ejde-39 4 8 saddle saddle NOUN ejde-39 4 9 - - PUNCT ejde-39 4 10 point point NOUN ejde-39 4 11 solutions solution NOUN ejde-39 4 12 of of ADP ejde-39 4 13 nonlinear nonlinear ADJ ejde-39 4 14 indefinite indefinite ADJ ejde-39 4 15 problems problem NOUN ejde-39 4 16 yavdat yavdat PROPN ejde-39 4 17 il’yasov il’yasov PROPN ejde-39 4 18 , , PUNCT ejde-39 4 19 edcarlos edcarlos PROPN ejde-39 4 20 d. d. PROPN ejde-39 4 21 silva silva PROPN ejde-39 4 22 , , PUNCT ejde-39 4 23 maxwell maxwell PROPN ejde-39 4 24 l. l. PROPN ejde-39 4 25 silva silva PROPN ejde-39 4 26 abstract abstract PROPN ejde-39 4 27 . . PUNCT ejde-39 5 1 a a DET ejde-39 5 2 minimax minimax NOUN ejde-39 5 3 variational variational ADJ ejde-39 5 4 method method NOUN ejde-39 5 5 for for ADP ejde-39 5 6 finding find VERB ejde-39 5 7 mountain mountain NOUN ejde-39 5 8 pass pass NOUN ejde-39 5 9 - - PUNCT ejde-39 5 10 type type NOUN ejde-39 5 11 solutions solution NOUN ejde-39 5 12 with with ADP ejde-39 5 13 prescribed prescribed ADJ ejde-39 5 14 energy energy NOUN ejde-39 5 15 levels level NOUN ejde-39 5 16 is be AUX ejde-39 5 17 introduced introduce VERB ejde-39 5 18 . . PUNCT ejde-39 6 1 the the DET ejde-39 6 2 method method NOUN ejde-39 6 3 is be AUX ejde-39 6 4 based base VERB ejde-39 6 5 on on ADP ejde-39 6 6 application application NOUN ejde-39 6 7 of of ADP ejde-39 6 8 the the DET ejde-39 6 9 linking linking NOUN ejde-39 6 10 theorem theorem NOUN ejde-39 6 11 to to ADP ejde-39 6 12 the the DET ejde-39 6 13 energy energy NOUN ejde-39 6 14 - - PUNCT ejde-39 6 15 level level NOUN ejde-39 6 16 nonlinear nonlinear ADJ ejde-39 6 17 rayleigh rayleigh PROPN ejde-39 6 18 quotients quotient NOUN ejde-39 6 19 which which PRON ejde-39 6 20 critical critical ADJ ejde-39 6 21 points point NOUN ejde-39 6 22 correspond correspond VERB ejde-39 6 23 to to ADP ejde-39 6 24 the the DET ejde-39 6 25 solutions solution NOUN ejde-39 6 26 of of ADP ejde-39 6 27 the the DET ejde-39 6 28 equation equation NOUN ejde-39 6 29 with with ADP ejde-39 6 30 prescribed prescribed ADJ ejde-39 6 31 energy energy NOUN ejde-39 6 32 . . PUNCT ejde-39 7 1 an an DET ejde-39 7 2 application application NOUN ejde-39 7 3 of of ADP ejde-39 7 4 the the DET ejde-39 7 5 method method NOUN ejde-39 7 6 to to PART ejde-39 7 7 nonlinear nonlinear ADJ ejde-39 7 8 indefinite indefinite ADJ ejde-39 7 9 elliptic elliptic ADJ ejde-39 7 10 problems problem NOUN ejde-39 7 11 with with ADP ejde-39 7 12 nonlinearities nonlinearitie NOUN ejde-39 7 13 that that PRON ejde-39 7 14 does do AUX ejde-39 7 15 not not PART ejde-39 7 16 satisfy satisfy VERB ejde-39 7 17 the the DET ejde-39 7 18 ambrosetti ambrosetti NOUN ejde-39 7 19 - - PUNCT ejde-39 7 20 rabinowitz rabinowitz NOUN ejde-39 7 21 growth growth NOUN ejde-39 7 22 conditions condition NOUN ejde-39 7 23 is be AUX ejde-39 7 24 also also ADV ejde-39 7 25 presented present VERB ejde-39 7 26 . . PUNCT ejde-39 8 1 1 1 X ejde-39 8 2 . . X ejde-39 8 3 introduction introduction NOUN ejde-39 8 4 let let VERB ejde-39 8 5 ω ω NOUN ejde-39 8 6 be be AUX ejde-39 8 7 a a DET ejde-39 8 8 bounded bounded ADJ ejde-39 8 9 smooth smooth ADJ ejde-39 8 10 domain domain NOUN ejde-39 8 11 in in ADP ejde-39 8 12 rn rn PROPN ejde-39 8 13 , , PUNCT ejde-39 8 14 n n CCONJ ejde-39 8 15 ≥ ≥ NOUN ejde-39 8 16 1 1 NUM ejde-39 8 17 and and CCONJ ejde-39 8 18 consider consider VERB ejde-39 8 19 −∆u− −∆u− PROPN ejde-39 8 20 λu λu NOUN ejde-39 8 21 = = NOUN ejde-39 8 22 µ|u|q−1u+ µ|u|q−1u+ NOUN ejde-39 8 23 g(x g(x NOUN ejde-39 8 24 , , PUNCT ejde-39 8 25 u u NOUN ejde-39 8 26 ) ) PUNCT ejde-39 8 27 in in ADP ejde-39 8 28 ω ω PROPN ejde-39 8 29 , , PUNCT ejde-39 8 30 u u NOUN ejde-39 8 31 = = NOUN ejde-39 8 32 0 0 NUM ejde-39 8 33 on on ADP ejde-39 8 34 ∂ω ∂ω PROPN ejde-39 8 35 , , PUNCT ejde-39 8 36 ( ( PUNCT ejde-39 8 37 1.1 1.1 NUM ejde-39 8 38 ) ) PUNCT ejde-39 8 39 where where SCONJ ejde-39 8 40 λ λ PROPN ejde-39 8 41 ∈ ∈ PROPN ejde-39 8 42 r r NOUN ejde-39 8 43 , , PUNCT ejde-39 8 44 µ µ X ejde-39 8 45 > > X ejde-39 8 46 0 0 NUM ejde-39 8 47 , , PUNCT ejde-39 8 48 1 1 NUM ejde-39 8 49 < < X ejde-39 8 50 q q X ejde-39 8 51 < < X ejde-39 8 52 2 2 NUM ejde-39 8 53 , , PUNCT ejde-39 8 54 g g NOUN ejde-39 8 55 : : PUNCT ejde-39 8 56 ω ω NUM ejde-39 8 57 × × NOUN ejde-39 8 58 r r NOUN ejde-39 8 59 → → SYM ejde-39 8 60 r r NOUN ejde-39 8 61 is be AUX ejde-39 8 62 a a DET ejde-39 8 63 carathéodory carathéodory NOUN ejde-39 8 64 function function NOUN ejde-39 8 65 with with ADP ejde-39 8 66 primitive primitive ADJ ejde-39 8 67 g(x g(x NOUN ejde-39 8 68 , , PUNCT ejde-39 8 69 u u NOUN ejde-39 8 70 ) ) PUNCT ejde-39 8 71 . . PUNCT ejde-39 9 1 the the DET ejde-39 9 2 problem problem NOUN ejde-39 9 3 has have VERB ejde-39 9 4 a a DET ejde-39 9 5 variational variational ADJ ejde-39 9 6 structure structure NOUN ejde-39 9 7 and and CCONJ ejde-39 9 8 under under ADP ejde-39 9 9 some some DET ejde-39 9 10 assumptions assumption NOUN ejde-39 9 11 ( ( PUNCT ejde-39 9 12 see see VERB ejde-39 9 13 below below ADV ejde-39 9 14 ( ( PUNCT ejde-39 9 15 a1 a1 NOUN ejde-39 9 16 ) ) PUNCT ejde-39 9 17 ) ) PUNCT ejde-39 10 1 the the DET ejde-39 10 2 associated associated ADJ ejde-39 10 3 energy energy NOUN ejde-39 10 4 functional functional ADJ ejde-39 10 5 eλ,µ eλ,µ X ejde-39 10 6 ∈ ∈ NOUN ejde-39 10 7 c1(ẘ c1(ẘ NOUN ejde-39 10 8 1 1 NUM ejde-39 10 9 2 2 NUM ejde-39 10 10 ( ( PUNCT ejde-39 10 11 ω),r ω),r NOUN ejde-39 10 12 ) ) PUNCT ejde-39 10 13 is be AUX ejde-39 10 14 eλ,µ(u eλ,µ(u NOUN ejde-39 10 15 ) ) PUNCT ejde-39 10 16 = = SYM ejde-39 10 17 1 1 NUM ejde-39 10 18 2 2 NUM ejde-39 10 19 ( ( PUNCT ejde-39 10 20 ∫ ∫ PROPN ejde-39 10 21 |∇u|2dx− |∇u|2dx− PUNCT ejde-39 10 22 λ λ PROPN ejde-39 10 23 ∫ ∫ PROPN ejde-39 10 24 |u|2dx |u|2dx PROPN ejde-39 10 25 ) ) PUNCT ejde-39 10 26 − − PROPN ejde-39 10 27 µ µ PRON ejde-39 10 28 q q X ejde-39 10 29 ∫ ∫ PROPN ejde-39 10 30 |u|qdx− |u|qdx− PROPN ejde-39 10 31 ∫ ∫ PROPN ejde-39 10 32 g(x g(x PROPN ejde-39 10 33 , , PUNCT ejde-39 10 34 u)dx u)dx PROPN ejde-39 10 35 . . PUNCT ejde-39 11 1 by by ADP ejde-39 11 2 definition definition NOUN ejde-39 11 3 , , PUNCT ejde-39 11 4 the the DET ejde-39 11 5 critical critical ADJ ejde-39 11 6 point point NOUN ejde-39 11 7 u u NOUN ejde-39 11 8 ∈ ∈ PROPN ejde-39 11 9 ẘ ẘ VERB ejde-39 11 10 1 1 NUM ejde-39 11 11 2 2 NUM ejde-39 11 12 ( ( PUNCT ejde-39 11 13 ω ω NOUN ejde-39 11 14 ) ) PUNCT ejde-39 11 15 of of ADP ejde-39 11 16 eλ,µ(u eλ,µ(u NOUN ejde-39 11 17 ) ) PUNCT ejde-39 11 18 is be AUX ejde-39 11 19 a a DET ejde-39 11 20 weak weak ADJ ejde-39 11 21 solution solution NOUN ejde-39 11 22 to to ADP ejde-39 11 23 ( ( PUNCT ejde-39 11 24 1.1 1.1 NUM ejde-39 11 25 ) ) PUNCT ejde-39 11 26 . . PUNCT ejde-39 12 1 the the DET ejde-39 12 2 problem problem NOUN ejde-39 12 3 with with ADP ejde-39 12 4 λ λ PROPN ejde-39 12 5 > > X ejde-39 12 6 λ1 λ1 PROPN ejde-39 12 7 , , PUNCT ejde-39 12 8 where where SCONJ ejde-39 12 9 λ1 λ1 PROPN ejde-39 12 10 is be AUX ejde-39 12 11 the the DET ejde-39 12 12 principal principal ADJ ejde-39 12 13 eigenvalue eigenvalue NOUN ejde-39 12 14 of of ADP ejde-39 12 15 the the DET ejde-39 12 16 operator operator NOUN ejde-39 12 17 ( ( PUNCT ejde-39 12 18 −∆ −∆ NOUN ejde-39 12 19 ) ) PUNCT ejde-39 12 20 in in ADP ejde-39 12 21 ẘ ẘ VERB ejde-39 12 22 1 1 NUM ejde-39 12 23 2 2 NUM ejde-39 12 24 ( ( PUNCT ejde-39 12 25 ω ω NOUN ejde-39 12 26 ) ) PUNCT ejde-39 12 27 is be AUX ejde-39 12 28 called call VERB ejde-39 12 29 indefinite indefinite ADJ ejde-39 12 30 due due ADP ejde-39 12 31 to to ADP ejde-39 12 32 the the DET ejde-39 12 33 fact fact NOUN ejde-39 12 34 that that SCONJ ejde-39 12 35 the the DET ejde-39 12 36 linear linear ADJ ejde-39 12 37 part part NOUN ejde-39 12 38 of of ADP ejde-39 12 39 ( ( PUNCT ejde-39 12 40 1.1 1.1 NUM ejde-39 12 41 ) ) PUNCT ejde-39 12 42 is be AUX ejde-39 12 43 indefinite indefinite ADJ ejde-39 12 44 ( ( PUNCT ejde-39 12 45 see see VERB ejde-39 12 46 [ [ X ejde-39 12 47 5 5 NUM ejde-39 12 48 , , PUNCT ejde-39 12 49 25 25 NUM ejde-39 12 50 ] ] PUNCT ejde-39 12 51 ) ) PUNCT ejde-39 12 52 . . PUNCT ejde-39 13 1 equation equation NOUN ejde-39 13 2 ( ( PUNCT ejde-39 13 3 1.1 1.1 NUM ejde-39 13 4 ) ) PUNCT ejde-39 13 5 is be AUX ejde-39 13 6 related relate VERB ejde-39 13 7 to to ADP ejde-39 13 8 finding find VERB ejde-39 13 9 the the DET ejde-39 13 10 amplitude amplitude NOUN ejde-39 13 11 function function NOUN ejde-39 13 12 u u NOUN ejde-39 13 13 of of ADP ejde-39 13 14 the the DET ejde-39 13 15 standing stand VERB ejde-39 13 16 waves wave NOUN ejde-39 13 17 ψ ψ X ejde-39 13 18 = = NOUN ejde-39 13 19 eiλtu eiλtu NOUN ejde-39 13 20 to to ADP ejde-39 13 21 the the DET ejde-39 13 22 nonlinear nonlinear ADJ ejde-39 13 23 schrödinger schrödinger NOUN ejde-39 13 24 ( ( PUNCT ejde-39 13 25 nls nls NOUN ejde-39 13 26 ) ) PUNCT ejde-39 13 27 equation equation NOUN ejde-39 13 28 iψt iψt NOUN ejde-39 13 29 = = PUNCT ejde-39 14 1 ∆ψ ∆ψ PROPN ejde-39 14 2 + + PUNCT ejde-39 14 3 µ|ψ|q−2ψ µ|ψ|q−2ψ PROPN ejde-39 15 1 + + CCONJ ejde-39 15 2 g(x g(x NOUN ejde-39 15 3 , , PUNCT ejde-39 15 4 ψ ψ NOUN ejde-39 15 5 ) ) PUNCT ejde-39 15 6 , , PUNCT ejde-39 15 7 ( ( PUNCT ejde-39 15 8 t t PROPN ejde-39 15 9 , , PUNCT ejde-39 15 10 x x X ejde-39 15 11 ) ) PUNCT ejde-39 15 12 ∈ ∈ PROPN ejde-39 15 13 r+ r+ PUNCT ejde-39 15 14 × × PROPN ejde-39 15 15 ω ω PROPN ejde-39 15 16 , , PUNCT ejde-39 15 17 ( ( PUNCT ejde-39 15 18 1.2 1.2 NUM ejde-39 15 19 ) ) PUNCT ejde-39 15 20 where where SCONJ ejde-39 15 21 ψ ψ NOUN ejde-39 15 22 is be AUX ejde-39 15 23 a a DET ejde-39 15 24 complex complex ADV ejde-39 15 25 - - PUNCT ejde-39 15 26 valued value VERB ejde-39 15 27 function function NOUN ejde-39 15 28 of of ADP ejde-39 15 29 ( ( PUNCT ejde-39 15 30 t t PROPN ejde-39 15 31 , , PUNCT ejde-39 15 32 x x NOUN ejde-39 15 33 ) ) PUNCT ejde-39 15 34 , , PUNCT ejde-39 15 35 and and CCONJ ejde-39 15 36 it it PRON ejde-39 15 37 is be AUX ejde-39 15 38 supposed suppose VERB ejde-39 15 39 that that SCONJ ejde-39 15 40 g(x g(x NOUN ejde-39 15 41 , , PUNCT ejde-39 15 42 ρeiθ ρeiθ NUM ejde-39 15 43 ) ) PUNCT ejde-39 16 1 = = SYM ejde-39 16 2 g(x g(x NOUN ejde-39 16 3 , , PUNCT ejde-39 16 4 ρ)eiθ ρ)eiθ PROPN ejde-39 16 5 a.e a.e PROPN ejde-39 16 6 . . PROPN ejde-39 16 7 ω ω PROPN ejde-39 16 8 , , PUNCT ejde-39 16 9 for for ADP ejde-39 16 10 all all DET ejde-39 16 11 ρ ρ NOUN ejde-39 16 12 , , PUNCT ejde-39 16 13 θ θ PROPN ejde-39 16 14 ∈ ∈ PROPN ejde-39 16 15 r. r. NOUN ejde-39 16 16 the the DET ejde-39 16 17 cauchy cauchy PROPN ejde-39 16 18 problem problem NOUN ejde-39 16 19 for for ADP ejde-39 16 20 ( ( PUNCT ejde-39 16 21 1.2 1.2 NUM ejde-39 16 22 ) ) PUNCT ejde-39 16 23 with with ADP ejde-39 16 24 the the DET ejde-39 16 25 initial initial ADJ ejde-39 16 26 value value NOUN ejde-39 16 27 ψ0 ψ0 NOUN ejde-39 16 28 ∈ ∈ PROPN ejde-39 16 29 ẘ ẘ VERB ejde-39 16 30 1 1 NUM ejde-39 16 31 2 2 NUM ejde-39 16 32 ( ( PUNCT ejde-39 16 33 ω ω NOUN ejde-39 16 34 ) ) PUNCT ejde-39 16 35 is be AUX ejde-39 16 36 locally locally ADV ejde-39 16 37 well well ADV ejde-39 16 38 posed pose VERB ejde-39 16 39 and and CCONJ ejde-39 16 40 for for ADP ejde-39 16 41 some some DET ejde-39 16 42 t t NOUN ejde-39 16 43 ( ( PUNCT ejde-39 16 44 ψ0 ψ0 PROPN ejde-39 16 45 ) ) PUNCT ejde-39 16 46 > > X ejde-39 16 47 0 0 PUNCT ejde-39 16 48 has have VERB ejde-39 16 49 a a DET ejde-39 16 50 unique unique ADJ ejde-39 16 51 2020 2020 NUM ejde-39 16 52 mathematics mathematic NOUN ejde-39 16 53 subject subject ADJ ejde-39 16 54 classification classification NOUN ejde-39 16 55 . . PUNCT ejde-39 17 1 35g15 35g15 NUM ejde-39 17 2 , , PUNCT ejde-39 17 3 35g20 35g20 NUM ejde-39 17 4 , , PUNCT ejde-39 17 5 35g25 35g25 NUM ejde-39 17 6 , , PUNCT ejde-39 17 7 35g30 35g30 NUM ejde-39 17 8 . . PUNCT ejde-39 18 1 key key ADJ ejde-39 18 2 words word NOUN ejde-39 18 3 and and CCONJ ejde-39 18 4 phrases phrase NOUN ejde-39 18 5 . . PUNCT ejde-39 19 1 indefinite indefinite ADJ ejde-39 19 2 problems problem NOUN ejde-39 19 3 ; ; PUNCT ejde-39 19 4 linking link VERB ejde-39 19 5 theorems theorem NOUN ejde-39 19 6 ; ; PUNCT ejde-39 19 7 rayleigh rayleigh PROPN ejde-39 19 8 quotient quotient NOUN ejde-39 19 9 . . PUNCT ejde-39 20 1 © © ADP ejde-39 20 2 2023 2023 NUM ejde-39 20 3 . . PUNCT ejde-39 21 1 this this DET ejde-39 21 2 work work NOUN ejde-39 21 3 is be AUX ejde-39 21 4 licensed license VERB ejde-39 21 5 under under ADP ejde-39 21 6 a a DET ejde-39 21 7 cc cc NOUN ejde-39 21 8 by by ADP ejde-39 21 9 4.0 4.0 NUM ejde-39 21 10 license license NOUN ejde-39 21 11 . . PUNCT ejde-39 22 1 submitted submit VERB ejde-39 22 2 december december PROPN ejde-39 22 3 5 5 NUM ejde-39 22 4 , , PUNCT ejde-39 22 5 2022 2022 NUM ejde-39 22 6 . . PUNCT ejde-39 23 1 published publish VERB ejde-39 23 2 march march PROPN ejde-39 23 3 4 4 NUM ejde-39 23 4 , , PUNCT ejde-39 23 5 2023 2023 NUM ejde-39 23 6 . . PUNCT ejde-39 24 1 1 1 NUM ejde-39 24 2 2 2 NUM ejde-39 24 3 y. y. NOUN ejde-39 24 4 il’yasov il’yasov PROPN ejde-39 24 5 , , PUNCT ejde-39 24 6 e. e. PROPN ejde-39 24 7 d. d. PROPN ejde-39 24 8 silva silva PROPN ejde-39 24 9 , , PUNCT ejde-39 24 10 m. m. NOUN ejde-39 24 11 l. l. PROPN ejde-39 24 12 silva silva PROPN ejde-39 24 13 ejde-2023/23 ejde-2023/23 PROPN ejde-39 24 14 local local ADJ ejde-39 24 15 solution solution NOUN ejde-39 24 16 ψ ψ ADP ejde-39 24 17 ∈ ∈ PROPN ejde-39 24 18 c([0 c([0 NOUN ejde-39 24 19 , , PUNCT ejde-39 24 20 t t PROPN ejde-39 24 21 ( ( PUNCT ejde-39 24 22 ψ0 ψ0 PROPN ejde-39 24 23 ) ) PUNCT ejde-39 24 24 ) ) PUNCT ejde-39 24 25 , , PUNCT ejde-39 24 26 ẘ ẘ VERB ejde-39 24 27 1 1 NUM ejde-39 24 28 2 2 NUM ejde-39 24 29 ( ( PUNCT ejde-39 24 30 ω ω NOUN ejde-39 24 31 ) ) PUNCT ejde-39 24 32 ) ) PUNCT ejde-39 24 33 ∩ ∩ NOUN ejde-39 24 34 c1([0 c1([0 PROPN ejde-39 24 35 , , PUNCT ejde-39 24 36 t t PROPN ejde-39 24 37 ( ( PUNCT ejde-39 24 38 ψ0 ψ0 PROPN ejde-39 24 39 ) ) PUNCT ejde-39 24 40 ) ) PUNCT ejde-39 24 41 , , PUNCT ejde-39 24 42 ẘ−1 ẘ−1 PROPN ejde-39 24 43 2 2 NUM ejde-39 24 44 ( ( PUNCT ejde-39 24 45 ω ω NOUN ejde-39 24 46 ) ) PUNCT ejde-39 24 47 ) ) PUNCT ejde-39 24 48 ( ( PUNCT ejde-39 24 49 see see VERB ejde-39 24 50 , , PUNCT ejde-39 24 51 e.g e.g PROPN ejde-39 24 52 , , PUNCT ejde-39 24 53 [ [ X ejde-39 24 54 10 10 NUM ejde-39 24 55 ] ] NUM ejde-39 24 56 ) ) PUNCT ejde-39 24 57 . . PUNCT ejde-39 25 1 moreover moreover ADV ejde-39 25 2 , , PUNCT ejde-39 25 3 it it PRON ejde-39 25 4 holds hold VERB ejde-39 25 5 energy energy NOUN ejde-39 25 6 and and CCONJ ejde-39 25 7 mass mass ADJ ejde-39 25 8 conservation conservation NOUN ejde-39 25 9 laws law NOUN ejde-39 25 10 : : PUNCT ejde-39 25 11 hµ(ψ(t hµ(ψ(t X ejde-39 25 12 ) ) PUNCT ejde-39 25 13 ) ) PUNCT ejde-39 25 14 : : PUNCT ejde-39 26 1 = = SYM ejde-39 26 2 ∫ ∫ PROPN ejde-39 26 3 ( ( PUNCT ejde-39 26 4 1 1 NUM ejde-39 26 5 2 2 NUM ejde-39 26 6 |∇ψ|2 |∇ψ|2 NOUN ejde-39 26 7 − − PROPN ejde-39 26 8 µ µ ADJ ejde-39 26 9 q q X ejde-39 26 10 |ψ|q |ψ|q NOUN ejde-39 26 11 −g(x −g(x PROPN ejde-39 26 12 , , PUNCT ejde-39 26 13 ψ ψ NOUN ejde-39 26 14 ) ) PUNCT ejde-39 26 15 ) ) PUNCT ejde-39 26 16 dx dx PROPN ejde-39 26 17 = = SYM ejde-39 26 18 const const PROPN ejde-39 26 19 , , PUNCT ejde-39 26 20 q(ψ(t q(ψ(t PROPN ejde-39 26 21 ) ) PUNCT ejde-39 26 22 ) ) PUNCT ejde-39 27 1 : : PUNCT ejde-39 27 2 = = SYM ejde-39 27 3 1 1 NUM ejde-39 27 4 2 2 NUM ejde-39 27 5 ∫ ∫ NOUN ejde-39 27 6 |ψ|2dx |ψ|2dx NUM ejde-39 27 7 = = SYM ejde-39 27 8 const const PROPN ejde-39 27 9 . . PUNCT ejde-39 28 1 as as ADP ejde-39 28 2 a a DET ejde-39 28 3 result result NOUN ejde-39 28 4 , , PUNCT ejde-39 28 5 the the DET ejde-39 28 6 energy energy NOUN ejde-39 28 7 functional functional ADJ ejde-39 28 8 ( ( PUNCT ejde-39 28 9 action action NOUN ejde-39 28 10 ) ) PUNCT ejde-39 28 11 eλ,µ(ψ(t eλ,µ(ψ(t PROPN ejde-39 28 12 ) ) PUNCT ejde-39 28 13 ) ) PUNCT ejde-39 29 1 : : PUNCT ejde-39 29 2 = = SYM ejde-39 29 3 hµ(ψ)− hµ(ψ)− X ejde-39 29 4 λq(ψ λq(ψ X ejde-39 29 5 ) ) PUNCT ejde-39 29 6 = = SYM ejde-39 30 1 const const ADJ ejde-39 30 2 , , PUNCT ejde-39 30 3 λ λ PROPN ejde-39 30 4 ∈ ∈ NOUN ejde-39 30 5 r r NOUN ejde-39 30 6 is be AUX ejde-39 30 7 also also ADV ejde-39 30 8 conserved conserve VERB ejde-39 30 9 . . PUNCT ejde-39 31 1 this this DET ejde-39 31 2 article article NOUN ejde-39 31 3 focuses focus VERB ejde-39 31 4 on on ADP ejde-39 31 5 the the DET ejde-39 31 6 existence existence NOUN ejde-39 31 7 of of ADP ejde-39 31 8 the the DET ejde-39 31 9 so so ADV ejde-39 31 10 - - PUNCT ejde-39 31 11 called call VERB ejde-39 31 12 prescribed prescribed ADJ ejde-39 31 13 energy energy NOUN ejde-39 31 14 solution solution NOUN ejde-39 31 15 of of ADP ejde-39 31 16 ( ( PUNCT ejde-39 31 17 1.1 1.1 NUM ejde-39 31 18 ) ) PUNCT ejde-39 31 19 , , PUNCT ejde-39 31 20 i.e. i.e. X ejde-39 31 21 , , PUNCT ejde-39 31 22 which which PRON ejde-39 31 23 for for ADP ejde-39 31 24 a a DET ejde-39 31 25 given give VERB ejde-39 31 26 energy energy NOUN ejde-39 31 27 e e NOUN ejde-39 31 28 ∈ ∈ NOUN ejde-39 31 29 r r NOUN ejde-39 31 30 satisfies satisfie NOUN ejde-39 31 31 eλ,µ(ue eλ,µ(ue NOUN ejde-39 31 32 ) ) PUNCT ejde-39 31 33 = = SYM ejde-39 31 34 e e X ejde-39 31 35 , , PUNCT ejde-39 31 36 deλ,µ(ue deλ,µ(ue NOUN ejde-39 31 37 ) ) PUNCT ejde-39 31 38 = = SYM ejde-39 31 39 0 0 NUM ejde-39 31 40 , , PUNCT ejde-39 31 41 where where SCONJ ejde-39 31 42 “ " PUNCT ejde-39 31 43 d d X ejde-39 31 44 ( ( PUNCT ejde-39 31 45 · · PUNCT ejde-39 31 46 ) ) PUNCT ejde-39 31 47 ” " PUNCT ejde-39 31 48 denotes denote VERB ejde-39 31 49 the the DET ejde-39 31 50 fréchet fréchet NOUN ejde-39 31 51 derivative derivative NOUN ejde-39 31 52 . . PUNCT ejde-39 32 1 in in ADP ejde-39 32 2 the the DET ejde-39 32 3 literature literature NOUN ejde-39 32 4 , , PUNCT ejde-39 32 5 solutions solution NOUN ejde-39 32 6 to to ADP ejde-39 32 7 the the DET ejde-39 32 8 schrödinger schrödinger NOUN ejde-39 32 9 equations equation NOUN ejde-39 32 10 having have VERB ejde-39 32 11 a a DET ejde-39 32 12 prescribed prescribed ADJ ejde-39 32 13 frequency frequency NOUN ejde-39 32 14 λ λ NOUN ejde-39 32 15 and and CCONJ ejde-39 32 16 unknowns unknown VERB ejde-39 32 17 energy energy NOUN ejde-39 32 18 e e NOUN ejde-39 32 19 and and CCONJ ejde-39 32 20 mass mass NOUN ejde-39 32 21 α α NOUN ejde-39 32 22 = = SYM ejde-39 32 23 q(u q(u NOUN ejde-39 32 24 ) ) PUNCT ejde-39 32 25 are be AUX ejde-39 32 26 commonly commonly ADV ejde-39 32 27 studied study VERB ejde-39 32 28 ( ( PUNCT ejde-39 32 29 see see VERB ejde-39 32 30 , , PUNCT ejde-39 32 31 e.g e.g NOUN ejde-39 32 32 , , PUNCT ejde-39 32 33 [ [ X ejde-39 32 34 10 10 NUM ejde-39 32 35 , , PUNCT ejde-39 32 36 26 26 NUM ejde-39 32 37 ] ] PUNCT ejde-39 32 38 ) ) PUNCT ejde-39 32 39 . . PUNCT ejde-39 33 1 an an DET ejde-39 33 2 alternative alternative ADJ ejde-39 33 3 formulation formulation NOUN ejde-39 33 4 which which PRON ejde-39 33 5 has have AUX ejde-39 33 6 also also ADV ejde-39 33 7 been be AUX ejde-39 33 8 actively actively ADV ejde-39 33 9 investigated investigate VERB ejde-39 33 10 over over ADP ejde-39 33 11 the the DET ejde-39 33 12 last last ADJ ejde-39 33 13 decades decade NOUN ejde-39 33 14 consists consist VERB ejde-39 33 15 of of ADP ejde-39 33 16 finding find VERB ejde-39 33 17 the the DET ejde-39 33 18 solution solution NOUN ejde-39 33 19 u u INTJ ejde-39 33 20 to to ADP ejde-39 33 21 ( ( PUNCT ejde-39 33 22 1.2 1.2 NUM ejde-39 33 23 ) ) PUNCT ejde-39 33 24 having having AUX ejde-39 33 25 prescribed prescribe VERB ejde-39 33 26 mass mass NOUN ejde-39 33 27 α α NOUN ejde-39 33 28 , , PUNCT ejde-39 33 29 while while SCONJ ejde-39 33 30 λ λ PROPN ejde-39 33 31 and and CCONJ ejde-39 33 32 e e PROPN ejde-39 33 33 are be AUX ejde-39 33 34 unknown unknown ADJ ejde-39 33 35 ( ( PUNCT ejde-39 33 36 see see VERB ejde-39 33 37 , , PUNCT ejde-39 33 38 e.g. e.g. ADV ejde-39 33 39 , , PUNCT ejde-39 33 40 [ [ X ejde-39 33 41 4 4 NUM ejde-39 33 42 , , PUNCT ejde-39 33 43 11 11 NUM ejde-39 33 44 , , PUNCT ejde-39 33 45 22 22 NUM ejde-39 33 46 , , PUNCT ejde-39 33 47 27 27 NUM ejde-39 33 48 ] ] NUM ejde-39 33 49 ) ) PUNCT ejde-39 33 50 . . PUNCT ejde-39 34 1 mathematically mathematically ADV ejde-39 34 2 , , PUNCT ejde-39 34 3 all all DET ejde-39 34 4 three three NUM ejde-39 34 5 approaches approach NOUN ejde-39 34 6 , , PUNCT ejde-39 34 7 namely namely ADV ejde-39 34 8 , , PUNCT ejde-39 34 9 prescribed prescribed ADJ ejde-39 34 10 frequency frequency NOUN ejde-39 34 11 , , PUNCT ejde-39 34 12 prescribed prescribed ADJ ejde-39 34 13 energy energy NOUN ejde-39 34 14 , , PUNCT ejde-39 34 15 and and CCONJ ejde-39 34 16 prescribed prescribe VERB ejde-39 34 17 mass mass NOUN ejde-39 34 18 , , PUNCT ejde-39 34 19 are be AUX ejde-39 34 20 equally equally ADV ejde-39 34 21 valid valid ADJ ejde-39 34 22 . . PUNCT ejde-39 35 1 moreover moreover ADV ejde-39 35 2 , , PUNCT ejde-39 35 3 all all PRON ejde-39 35 4 of of ADP ejde-39 35 5 these these DET ejde-39 35 6 approaches approach NOUN ejde-39 35 7 evidently evidently ADV ejde-39 35 8 are be AUX ejde-39 35 9 relevant relevant ADJ ejde-39 35 10 from from ADP ejde-39 35 11 the the DET ejde-39 35 12 physical physical ADJ ejde-39 35 13 point point NOUN ejde-39 35 14 of of ADP ejde-39 35 15 view view NOUN ejde-39 35 16 . . PUNCT ejde-39 36 1 in in ADP ejde-39 36 2 particular particular ADJ ejde-39 36 3 , , PUNCT ejde-39 36 4 the the DET ejde-39 36 5 approach approach NOUN ejde-39 36 6 with with ADP ejde-39 36 7 prescribed prescribed ADJ ejde-39 36 8 energy energy NOUN ejde-39 36 9 arises arise VERB ejde-39 36 10 in in ADP ejde-39 36 11 the the DET ejde-39 36 12 study study NOUN ejde-39 36 13 of of ADP ejde-39 36 14 inverse inverse NOUN ejde-39 36 15 problems problem NOUN ejde-39 36 16 and and CCONJ ejde-39 36 17 the the DET ejde-39 36 18 spectral spectral ADJ ejde-39 36 19 and and CCONJ ejde-39 36 20 scattering scatter VERB ejde-39 36 21 control control NOUN ejde-39 36 22 problems problem NOUN ejde-39 36 23 ( ( PUNCT ejde-39 36 24 see see VERB ejde-39 36 25 , , PUNCT ejde-39 36 26 e.g. e.g. ADV ejde-39 36 27 , , PUNCT ejde-39 36 28 [ [ X ejde-39 36 29 2 2 NUM ejde-39 36 30 , , PUNCT ejde-39 36 31 12 12 NUM ejde-39 36 32 , , PUNCT ejde-39 36 33 20 20 NUM ejde-39 36 34 , , PUNCT ejde-39 36 35 23 23 NUM ejde-39 36 36 , , PUNCT ejde-39 36 37 24 24 NUM ejde-39 36 38 , , PUNCT ejde-39 36 39 29 29 NUM ejde-39 36 40 ] ] PUNCT ejde-39 36 41 ) ) PUNCT ejde-39 36 42 . . PUNCT ejde-39 37 1 the the DET ejde-39 37 2 prescribed prescribed ADJ ejde-39 37 3 energy energy NOUN ejde-39 37 4 solutions solution NOUN ejde-39 37 5 of of ADP ejde-39 37 6 nonlinear nonlinear ADJ ejde-39 37 7 problems problem NOUN ejde-39 37 8 was be AUX ejde-39 37 9 studied study VERB ejde-39 37 10 recently recently ADV ejde-39 37 11 in in ADP ejde-39 37 12 [ [ X ejde-39 37 13 7 7 NUM ejde-39 37 14 , , PUNCT ejde-39 37 15 18 18 NUM ejde-39 37 16 , , PUNCT ejde-39 37 17 19 19 NUM ejde-39 37 18 ] ] PUNCT ejde-39 37 19 by by ADP ejde-39 37 20 using use VERB ejde-39 37 21 the the DET ejde-39 37 22 nonlinear nonlinear ADJ ejde-39 37 23 rayleigh rayleigh PROPN ejde-39 37 24 quotients quotient NOUN ejde-39 37 25 [ [ X ejde-39 37 26 17 17 NUM ejde-39 37 27 ] ] PUNCT ejde-39 37 28 . . PUNCT ejde-39 38 1 the the DET ejde-39 38 2 nonlinear nonlinear ADJ ejde-39 38 3 rayleigh rayleigh PROPN ejde-39 38 4 quotients quotient NOUN ejde-39 38 5 have have VERB ejde-39 38 6 the the DET ejde-39 38 7 remarkable remarkable ADJ ejde-39 38 8 property property NOUN ejde-39 38 9 that that PRON ejde-39 38 10 the the DET ejde-39 38 11 critical critical ADJ ejde-39 38 12 points point NOUN ejde-39 38 13 of of ADP ejde-39 38 14 these these DET ejde-39 38 15 functionals functional NOUN ejde-39 38 16 correspond correspond VERB ejde-39 38 17 to to ADP ejde-39 38 18 the the DET ejde-39 38 19 solutions solution NOUN ejde-39 38 20 of of ADP ejde-39 38 21 the the DET ejde-39 38 22 equations equation NOUN ejde-39 38 23 while while SCONJ ejde-39 38 24 having have VERB ejde-39 38 25 a a DET ejde-39 38 26 simpler simple ADJ ejde-39 38 27 structure structure NOUN ejde-39 38 28 than than ADP ejde-39 38 29 the the DET ejde-39 38 30 corresponding correspond VERB ejde-39 38 31 energy energy NOUN ejde-39 38 32 functionals functional NOUN ejde-39 38 33 ( ( PUNCT ejde-39 38 34 see see VERB ejde-39 38 35 , , PUNCT ejde-39 38 36 e.g. e.g. ADV ejde-39 38 37 , , PUNCT ejde-39 38 38 [ [ X ejde-39 38 39 17 17 NUM ejde-39 38 40 ] ] NUM ejde-39 38 41 ) ) PUNCT ejde-39 38 42 . . PUNCT ejde-39 39 1 they they PRON ejde-39 39 2 were be AUX ejde-39 39 3 particularly particularly ADV ejde-39 39 4 useful useful ADJ ejde-39 39 5 ( ( PUNCT ejde-39 39 6 see see NOUN ejde-39 39 7 , , PUNCT ejde-39 39 8 e.g. e.g. ADV ejde-39 39 9 , , PUNCT ejde-39 39 10 [ [ X ejde-39 39 11 17 17 NUM ejde-39 39 12 , , PUNCT ejde-39 39 13 19 19 NUM ejde-39 39 14 ] ] PUNCT ejde-39 39 15 ) ) PUNCT ejde-39 39 16 for for ADP ejde-39 39 17 finding find VERB ejde-39 39 18 nonnegative nonnegative ADJ ejde-39 39 19 solutions solution NOUN ejde-39 39 20 to to ADP ejde-39 39 21 zero zero NUM ejde-39 39 22 - - PUNCT ejde-39 39 23 mass mass NOUN ejde-39 39 24 problems problem NOUN ejde-39 39 25 [ [ X ejde-39 39 26 6 6 NUM ejde-39 39 27 ] ] PUNCT ejde-39 39 28 and and CCONJ ejde-39 39 29 detecting detect VERB ejde-39 39 30 s s NOUN ejde-39 39 31 - - PUNCT ejde-39 39 32 shaped shape VERB ejde-39 39 33 bifurcations bifurcation NOUN ejde-39 39 34 of of ADP ejde-39 39 35 nonlinear nonlinear ADJ ejde-39 39 36 partial partial ADJ ejde-39 39 37 differential differential NOUN ejde-39 39 38 equations equation NOUN ejde-39 39 39 [ [ X ejde-39 39 40 7 7 NUM ejde-39 39 41 ] ] PUNCT ejde-39 39 42 . . PUNCT ejde-39 40 1 the the DET ejde-39 40 2 nonlinear nonlinear ADJ ejde-39 40 3 rayleigh rayleigh PROPN ejde-39 40 4 quotients quotient NOUN ejde-39 40 5 method method NOUN ejde-39 40 6 and and CCONJ ejde-39 40 7 solutions solution NOUN ejde-39 40 8 with with ADP ejde-39 40 9 prescribed prescribed ADJ ejde-39 40 10 energies energy NOUN ejde-39 40 11 were be AUX ejde-39 40 12 used use VERB ejde-39 40 13 to to PART ejde-39 40 14 introduce introduce VERB ejde-39 40 15 a a DET ejde-39 40 16 generalization generalization NOUN ejde-39 40 17 of of ADP ejde-39 40 18 the the DET ejde-39 40 19 poincaré poincaré ADJ ejde-39 40 20 and and CCONJ ejde-39 40 21 courant courant ADJ ejde-39 40 22 - - PUNCT ejde-39 40 23 fischer fischer PROPN ejde-39 40 24 - - PUNCT ejde-39 40 25 weil weil PROPN ejde-39 40 26 minimization minimization NOUN ejde-39 40 27 principles principle NOUN ejde-39 40 28 to to ADP ejde-39 40 29 nonlinear nonlinear ADJ ejde-39 40 30 problems problem NOUN ejde-39 40 31 [ [ X ejde-39 40 32 18 18 NUM ejde-39 40 33 ] ] X ejde-39 40 34 , , PUNCT ejde-39 40 35 as as ADV ejde-39 40 36 well well ADV ejde-39 40 37 as as ADP ejde-39 40 38 to to PART ejde-39 40 39 study study VERB ejde-39 40 40 the the DET ejde-39 40 41 orbital orbital ADJ ejde-39 40 42 stability stability NOUN ejde-39 40 43 for for ADP ejde-39 40 44 ground ground NOUN ejde-39 40 45 states state NOUN ejde-39 40 46 of of ADP ejde-39 40 47 the the DET ejde-39 40 48 nls nls NOUN ejde-39 40 49 equations equation NOUN ejde-39 40 50 [ [ X ejde-39 40 51 7 7 NUM ejde-39 40 52 ] ] PUNCT ejde-39 40 53 . . PUNCT ejde-39 41 1 there there PRON ejde-39 41 2 are be VERB ejde-39 41 3 at at ADV ejde-39 41 4 least least ADJ ejde-39 41 5 two two NUM ejde-39 41 6 motivations motivation NOUN ejde-39 41 7 to to PART ejde-39 41 8 study study VERB ejde-39 41 9 prescribed prescribe VERB ejde-39 41 10 energy energy NOUN ejde-39 41 11 solutions solution NOUN ejde-39 41 12 of of ADP ejde-39 41 13 ( ( PUNCT ejde-39 41 14 1.1 1.1 NUM ejde-39 41 15 ) ) PUNCT ejde-39 41 16 , , PUNCT ejde-39 41 17 apart apart ADV ejde-39 41 18 from from ADP ejde-39 41 19 the the DET ejde-39 41 20 fact fact NOUN ejde-39 41 21 that that SCONJ ejde-39 41 22 it it PRON ejde-39 41 23 appears appear VERB ejde-39 41 24 in in ADP ejde-39 41 25 some some DET ejde-39 41 26 physical physical ADJ ejde-39 41 27 models model NOUN ejde-39 41 28 . . PUNCT ejde-39 42 1 first first ADV ejde-39 42 2 , , PUNCT ejde-39 42 3 we we PRON ejde-39 42 4 develop develop VERB ejde-39 42 5 the the DET ejde-39 42 6 nonlinear nonlinear ADJ ejde-39 42 7 rayleigh rayleigh PROPN ejde-39 42 8 quotient quotient NOUN ejde-39 42 9 method method NOUN ejde-39 42 10 for for ADP ejde-39 42 11 new new ADJ ejde-39 42 12 classes class NOUN ejde-39 42 13 of of ADP ejde-39 42 14 problems problem NOUN ejde-39 42 15 , , PUNCT ejde-39 42 16 in in ADP ejde-39 42 17 particular particular ADJ ejde-39 42 18 for for ADP ejde-39 42 19 equations equation NOUN ejde-39 42 20 with with ADP ejde-39 42 21 inhomogeneous inhomogeneous ADJ ejde-39 42 22 and and CCONJ ejde-39 42 23 general general ADJ ejde-39 42 24 forms form NOUN ejde-39 42 25 of of ADP ejde-39 42 26 nonlinearities nonlinearitie NOUN ejde-39 42 27 . . PUNCT ejde-39 43 1 second second ADJ ejde-39 43 2 , , PUNCT ejde-39 43 3 we we PRON ejde-39 43 4 develop develop VERB ejde-39 43 5 the the DET ejde-39 43 6 mountain mountain NOUN ejde-39 43 7 pass pass NOUN ejde-39 43 8 methods method NOUN ejde-39 43 9 in in ADP ejde-39 43 10 order order NOUN ejde-39 43 11 to to PART ejde-39 43 12 capture capture VERB ejde-39 43 13 qualitative qualitative ADJ ejde-39 43 14 properties property NOUN ejde-39 43 15 of of ADP ejde-39 43 16 the the DET ejde-39 43 17 solutions solution NOUN ejde-39 43 18 that that PRON ejde-39 43 19 it it PRON ejde-39 43 20 generates generate VERB ejde-39 43 21 . . PUNCT ejde-39 44 1 the the DET ejde-39 44 2 mountain mountain NOUN ejde-39 44 3 pass pass NOUN ejde-39 44 4 theorem theorem NOUN ejde-39 44 5 introduced introduce VERB ejde-39 44 6 by by ADP ejde-39 44 7 ambrosetti ambrosetti NOUN ejde-39 44 8 and and CCONJ ejde-39 44 9 rabinowitz rabinowitz VERB ejde-39 44 10 [ [ X ejde-39 44 11 1 1 NUM ejde-39 44 12 ] ] PUNCT ejde-39 44 13 and and CCONJ ejde-39 44 14 its its PRON ejde-39 44 15 generalization generalization NOUN ejde-39 44 16 as as ADP ejde-39 44 17 the the DET ejde-39 44 18 benci benci PROPN ejde-39 44 19 - - PUNCT ejde-39 44 20 rabinowitz rabinowitz PROPN ejde-39 44 21 linking linking NOUN ejde-39 44 22 theorem theorem NOUN ejde-39 44 23 [ [ X ejde-39 44 24 5 5 NUM ejde-39 44 25 ] ] PUNCT ejde-39 44 26 is be AUX ejde-39 44 27 a a DET ejde-39 44 28 powerful powerful ADJ ejde-39 44 29 tool tool NOUN ejde-39 44 30 to to PART ejde-39 44 31 establish establish VERB ejde-39 44 32 the the DET ejde-39 44 33 existence existence NOUN ejde-39 44 34 of of ADP ejde-39 44 35 solutions solution NOUN ejde-39 44 36 for for ADP ejde-39 44 37 nonlinear nonlinear ADJ ejde-39 44 38 problems problem NOUN ejde-39 44 39 of of ADP ejde-39 44 40 the the DET ejde-39 44 41 variational variational ADJ ejde-39 44 42 form form NOUN ejde-39 44 43 . . PUNCT ejde-39 45 1 the the DET ejde-39 45 2 solutions solution NOUN ejde-39 45 3 obtained obtain VERB ejde-39 45 4 by by ADP ejde-39 45 5 this this DET ejde-39 45 6 method method NOUN ejde-39 45 7 usually usually ADV ejde-39 45 8 correspond correspond VERB ejde-39 45 9 to to PART ejde-39 45 10 saddle saddle VERB ejde-39 45 11 critical critical ADJ ejde-39 45 12 points point NOUN ejde-39 45 13 of of ADP ejde-39 45 14 the the DET ejde-39 45 15 energy energy NOUN ejde-39 45 16 functional functional ADJ ejde-39 45 17 and and CCONJ ejde-39 45 18 are be AUX ejde-39 45 19 often often ADV ejde-39 45 20 referred refer VERB ejde-39 45 21 to to ADP ejde-39 45 22 as as ADP ejde-39 45 23 mountain mountain NOUN ejde-39 45 24 pass pass NOUN ejde-39 45 25 - - PUNCT ejde-39 45 26 type type NOUN ejde-39 45 27 solutions solution NOUN ejde-39 45 28 or or CCONJ ejde-39 45 29 saddle saddle NOUN ejde-39 45 30 - - PUNCT ejde-39 45 31 point point NOUN ejde-39 45 32 solutions solution NOUN ejde-39 45 33 . . PUNCT ejde-39 46 1 in in ADP ejde-39 46 2 essence essence NOUN ejde-39 46 3 , , PUNCT ejde-39 46 4 this this DET ejde-39 46 5 method method NOUN ejde-39 46 6 is be AUX ejde-39 46 7 topological topological ADJ ejde-39 46 8 , , PUNCT ejde-39 46 9 which which PRON ejde-39 46 10 makes make VERB ejde-39 46 11 it it PRON ejde-39 46 12 possible possible ADJ ejde-39 46 13 to to PART ejde-39 46 14 use use VERB ejde-39 46 15 it it PRON ejde-39 46 16 for for ADP ejde-39 46 17 solving solve VERB ejde-39 46 18 problems problem NOUN ejde-39 46 19 of of ADP ejde-39 46 20 very very ADV ejde-39 46 21 general general ADJ ejde-39 46 22 forms form NOUN ejde-39 46 23 . . PUNCT ejde-39 47 1 on on ADP ejde-39 47 2 the the DET ejde-39 47 3 other other ADJ ejde-39 47 4 hand hand NOUN ejde-39 47 5 , , PUNCT ejde-39 47 6 this this DET ejde-39 47 7 generality generality NOUN ejde-39 47 8 often often ADV ejde-39 47 9 makes make VERB ejde-39 47 10 it it PRON ejde-39 47 11 difficult difficult ADJ ejde-39 47 12 to to PART ejde-39 47 13 find find VERB ejde-39 47 14 out out ADP ejde-39 47 15 detailed detailed ADJ ejde-39 47 16 information information NOUN ejde-39 47 17 about about ADP ejde-39 47 18 the the DET ejde-39 47 19 ejde-2023/23 ejde-2023/23 NOUN ejde-39 47 20 prescribed prescribe VERB ejde-39 47 21 energy energy NOUN ejde-39 47 22 saddle saddle NOUN ejde-39 47 23 - - PUNCT ejde-39 47 24 point point NOUN ejde-39 47 25 solutions solution NOUN ejde-39 47 26 3 3 NUM ejde-39 47 27 obtained obtain VERB ejde-39 47 28 solutions solution NOUN ejde-39 47 29 . . PUNCT ejde-39 48 1 the the DET ejde-39 48 2 aim aim NOUN ejde-39 48 3 of of ADP ejde-39 48 4 this this DET ejde-39 48 5 work work NOUN ejde-39 48 6 is be AUX ejde-39 48 7 to to PART ejde-39 48 8 show show VERB ejde-39 48 9 that that SCONJ ejde-39 48 10 the the DET ejde-39 48 11 nonlinear nonlinear PROPN ejde-39 48 12 rayleigh rayleigh PROPN ejde-39 48 13 quotient quotient NOUN ejde-39 48 14 method method NOUN ejde-39 48 15 can can AUX ejde-39 48 16 be be AUX ejde-39 48 17 applied apply VERB ejde-39 48 18 to to PART ejde-39 48 19 generate generate VERB ejde-39 48 20 saddle saddle NOUN ejde-39 48 21 - - PUNCT ejde-39 48 22 point point NOUN ejde-39 48 23 solutions solution NOUN ejde-39 48 24 with with ADP ejde-39 48 25 prescribed prescribed ADJ ejde-39 48 26 energy energy NOUN ejde-39 48 27 within within ADP ejde-39 48 28 the the DET ejde-39 48 29 framework framework NOUN ejde-39 48 30 of of ADP ejde-39 48 31 the the DET ejde-39 48 32 linking linking NOUN ejde-39 48 33 theorem theorem VERB ejde-39 48 34 . . PUNCT ejde-39 49 1 let let VERB ejde-39 49 2 us we PRON ejde-39 49 3 state state VERB ejde-39 49 4 our our PRON ejde-39 49 5 main main ADJ ejde-39 49 6 result result NOUN ejde-39 49 7 . . PUNCT ejde-39 50 1 we we PRON ejde-39 50 2 seek seek VERB ejde-39 50 3 for for ADP ejde-39 50 4 prescribed prescribe VERB ejde-39 50 5 energy energy NOUN ejde-39 50 6 solutions solution NOUN ejde-39 50 7 using use VERB ejde-39 50 8 the the DET ejde-39 50 9 energy energy NOUN ejde-39 50 10 level level NOUN ejde-39 50 11 nonlinear nonlinear PROPN ejde-39 50 12 rayleigh rayleigh PROPN ejde-39 50 13 quotient quotient NOUN ejde-39 51 1 [ [ X ejde-39 51 2 7 7 NUM ejde-39 51 3 , , PUNCT ejde-39 51 4 17 17 NUM ejde-39 51 5 , , PUNCT ejde-39 51 6 19 19 NUM ejde-39 51 7 ] ] NOUN ejde-39 51 8 : : PUNCT ejde-39 51 9 reλ reλ NOUN ejde-39 51 10 ( ( PUNCT ejde-39 51 11 u u NOUN ejde-39 51 12 ) ) PUNCT ejde-39 51 13 : : PUNCT ejde-39 51 14 = = SYM ejde-39 51 15 1 1 NUM ejde-39 51 16 2 2 NUM ejde-39 51 17 ( ( PUNCT ejde-39 51 18 ∫ ∫ PROPN ejde-39 51 19 |∇u|2 |∇u|2 X ejde-39 51 20 dx− dx− PRON ejde-39 51 21 λ λ PROPN ejde-39 51 22 ∫ ∫ PROPN ejde-39 51 23 |u|2 |u|2 PROPN ejde-39 51 24 dx dx PROPN ejde-39 51 25 ) ) PUNCT ejde-39 51 26 − − ADP ejde-39 52 1 ∫ ∫ PROPN ejde-39 52 2 g(x g(x PROPN ejde-39 52 3 , , PUNCT ejde-39 52 4 u u NOUN ejde-39 52 5 ) ) PUNCT ejde-39 52 6 dx− dx− X ejde-39 52 7 e e PROPN ejde-39 52 8 1 1 NUM ejde-39 52 9 q q NOUN ejde-39 52 10 ∫ ∫ PROPN ejde-39 52 11 |u|qdx |u|qdx PROPN ejde-39 52 12 , , PUNCT ejde-39 52 13 for for ADP ejde-39 52 14 u u PROPN ejde-39 52 15 ∈ ∈ PROPN ejde-39 52 16 ẘ ẘ VERB ejde-39 52 17 1 1 NUM ejde-39 52 18 2 2 NUM ejde-39 52 19 ( ( PUNCT ejde-39 52 20 ω ω NOUN ejde-39 52 21 ) ) PUNCT ejde-39 52 22 \ \ NOUN ejde-39 52 23 { { PUNCT ejde-39 52 24 0 0 NUM ejde-39 52 25 } } PUNCT ejde-39 52 26 and and CCONJ ejde-39 52 27 e e PROPN ejde-39 52 28 ∈ ∈ PROPN ejde-39 52 29 r. r. PROPN ejde-39 52 30 notice notice VERB ejde-39 52 31 that that SCONJ ejde-39 52 32 for for ADP ejde-39 52 33 u u PROPN ejde-39 52 34 ∈ ∈ PROPN ejde-39 52 35 ẘ ẘ VERB ejde-39 52 36 1 1 NUM ejde-39 52 37 2 2 NUM ejde-39 52 38 ( ( PUNCT ejde-39 52 39 ω ω NOUN ejde-39 52 40 ) ) PUNCT ejde-39 52 41 \ \ NOUN ejde-39 52 42 { { PUNCT ejde-39 52 43 0 0 NUM ejde-39 52 44 } } PUNCT ejde-39 52 45 , , PUNCT ejde-39 52 46 λ λ X ejde-39 52 47 ∈ ∈ NOUN ejde-39 52 48 r r NOUN ejde-39 52 49 , , PUNCT ejde-39 52 50 and and CCONJ ejde-39 52 51 e e X ejde-39 52 52 ∈ ∈ PROPN ejde-39 52 53 r r NOUN ejde-39 52 54 , , PUNCT ejde-39 52 55 we we PRON ejde-39 52 56 have have VERB ejde-39 52 57 µ µ NOUN ejde-39 52 58 = = SYM ejde-39 52 59 reλ reλ NOUN ejde-39 52 60 ( ( PUNCT ejde-39 52 61 u u NOUN ejde-39 52 62 ) ) PUNCT ejde-39 52 63 ⇔ ⇔ PROPN ejde-39 52 64 eλ,µ(u eλ,µ(u NOUN ejde-39 52 65 ) ) PUNCT ejde-39 53 1 = = SYM ejde-39 53 2 e e NOUN ejde-39 53 3 , , PUNCT ejde-39 53 4 µ µ NOUN ejde-39 53 5 = = SYM ejde-39 53 6 reλ reλ NOUN ejde-39 53 7 ( ( PUNCT ejde-39 53 8 u u NOUN ejde-39 53 9 ) ) PUNCT ejde-39 53 10 , , PUNCT ejde-39 53 11 dre(u dre(u PROPN ejde-39 53 12 ) ) PUNCT ejde-39 53 13 = = SYM ejde-39 53 14 0 0 NUM ejde-39 53 15 ⇔ ⇔ PROPN ejde-39 53 16 eλ,µ(u eλ,µ(u NOUN ejde-39 53 17 ) ) PUNCT ejde-39 53 18 = = SYM ejde-39 53 19 e e NOUN ejde-39 53 20 , , PUNCT ejde-39 53 21 deλ,µ(u deλ,µ(u PROPN ejde-39 53 22 ) ) PUNCT ejde-39 54 1 = = PUNCT ejde-39 54 2 0 0 X ejde-39 54 3 . . PUNCT ejde-39 55 1 ( ( PUNCT ejde-39 55 2 1.3 1.3 NUM ejde-39 55 3 ) ) PUNCT ejde-39 55 4 we we PRON ejde-39 55 5 assume assume VERB ejde-39 55 6 that that SCONJ ejde-39 55 7 ( ( PUNCT ejde-39 55 8 a1 a1 NOUN ejde-39 55 9 ) ) PUNCT ejde-39 55 10 there there PRON ejde-39 55 11 exist exist VERB ejde-39 55 12 γ1 γ1 NOUN ejde-39 55 13 , , PUNCT ejde-39 55 14 γ2 γ2 PROPN ejde-39 55 15 ∈ ∈ PROPN ejde-39 55 16 ( ( PUNCT ejde-39 55 17 2 2 NUM ejde-39 55 18 , , PUNCT ejde-39 55 19 2∗ 2∗ NUM ejde-39 55 20 ) ) PUNCT ejde-39 55 21 , , PUNCT ejde-39 56 1 c c X ejde-39 56 2 > > X ejde-39 56 3 0 0 NUM ejde-39 57 1 such such ADJ ejde-39 57 2 that that SCONJ ejde-39 57 3 0 0 NUM ejde-39 57 4 ≤ ≤ NUM ejde-39 57 5 g(x g(x NOUN ejde-39 57 6 , , PUNCT ejde-39 57 7 u u NOUN ejde-39 57 8 ) ) PUNCT ejde-39 57 9 ≤ ≤ NOUN ejde-39 57 10 c(|u|γ1−1 c(|u|γ1−1 VERB ejde-39 57 11 + + CCONJ ejde-39 57 12 |u|γ2−1 |u|γ2−1 PROPN ejde-39 57 13 ) ) PUNCT ejde-39 57 14 a.e a.e PROPN ejde-39 57 15 . . PROPN ejde-39 57 16 ω ω PROPN ejde-39 57 17 , , PUNCT ejde-39 57 18 u u PROPN ejde-39 57 19 ∈ ∈ PROPN ejde-39 57 20 r r PROPN ejde-39 57 21 , , PUNCT ejde-39 57 22 ( ( PUNCT ejde-39 57 23 a2 a2 PROPN ejde-39 57 24 ) ) PUNCT ejde-39 57 25 there there PRON ejde-39 57 26 exist exist VERB ejde-39 57 27 α α PROPN ejde-39 57 28 > > X ejde-39 57 29 2 2 NUM ejde-39 57 30 , , PUNCT ejde-39 57 31 r0 r0 NOUN ejde-39 57 32 > > X ejde-39 57 33 0 0 NUM ejde-39 58 1 such such ADJ ejde-39 58 2 that that SCONJ ejde-39 58 3 αg(x αg(x NUM ejde-39 58 4 , , PUNCT ejde-39 58 5 u u NOUN ejde-39 58 6 ) ) PUNCT ejde-39 58 7 ≤ ≤ NOUN ejde-39 58 8 g(x g(x NOUN ejde-39 58 9 , , PUNCT ejde-39 58 10 u)u u)u ADJ ejde-39 58 11 a.e a.e PROPN ejde-39 58 12 . . PROPN ejde-39 58 13 ω ω PROPN ejde-39 58 14 , , PUNCT ejde-39 58 15 |u| |u| PROPN ejde-39 58 16 ≥ ≥ NOUN ejde-39 58 17 r0 r0 NOUN ejde-39 58 18 , , PUNCT ejde-39 58 19 where where SCONJ ejde-39 58 20 2∗ 2∗ NUM ejde-39 58 21 = = SYM ejde-39 58 22 2n/(n 2n/(n NUM ejde-39 58 23 − − NUM ejde-39 58 24 2 2 NUM ejde-39 58 25 ) ) PUNCT ejde-39 58 26 if if SCONJ ejde-39 58 27 n n PROPN ejde-39 58 28 > > X ejde-39 58 29 2 2 NUM ejde-39 58 30 , , PUNCT ejde-39 58 31 2∗ 2∗ NUM ejde-39 58 32 = = PUNCT ejde-39 59 1 + + NUM ejde-39 59 2 ∞ ∞ PROPN ejde-39 59 3 if if SCONJ ejde-39 59 4 n n PRON ejde-39 59 5 ≤ ≤ ADV ejde-39 59 6 2 2 NUM ejde-39 59 7 . . PUNCT ejde-39 60 1 the the DET ejde-39 60 2 operator operator NOUN ejde-39 60 3 ( ( PUNCT ejde-39 60 4 −∆ −∆ PROPN ejde-39 60 5 ) ) PUNCT ejde-39 60 6 with with ADP ejde-39 60 7 dirichlet dirichlet PROPN ejde-39 60 8 boundary boundary PROPN ejde-39 60 9 conditions condition NOUN ejde-39 60 10 defines define VERB ejde-39 60 11 a a DET ejde-39 60 12 self self NOUN ejde-39 60 13 - - PUNCT ejde-39 60 14 adjoint adjoint NOUN ejde-39 60 15 operator operator NOUN ejde-39 60 16 in in ADP ejde-39 60 17 l2(ω l2(ω PROPN ejde-39 60 18 ) ) PUNCT ejde-39 60 19 ( ( PUNCT ejde-39 60 20 see see VERB ejde-39 60 21 , , PUNCT ejde-39 60 22 e.g. e.g. ADV ejde-39 60 23 , , PUNCT ejde-39 60 24 [ [ X ejde-39 60 25 14 14 NUM ejde-39 60 26 ] ] PUNCT ejde-39 60 27 ) ) PUNCT ejde-39 60 28 and and CCONJ ejde-39 60 29 its its PRON ejde-39 60 30 spectrum spectrum NOUN ejde-39 60 31 consists consist VERB ejde-39 60 32 of of ADP ejde-39 60 33 an an DET ejde-39 60 34 infinite infinite ADJ ejde-39 60 35 sequence sequence NOUN ejde-39 60 36 ordered order VERB ejde-39 60 37 0 0 PUNCT ejde-39 60 38 < < X ejde-39 60 39 λ1 λ1 X ejde-39 60 40 < < X ejde-39 60 41 λ2 λ2 NOUN ejde-39 60 42 ≤ ≤ NOUN ejde-39 60 43 . . PUNCT ejde-39 60 44 . . PUNCT ejde-39 61 1 . . PUNCT ejde-39 62 1 of of ADP ejde-39 62 2 eigenvalues eigenvalue NOUN ejde-39 62 3 repeated repeat VERB ejde-39 62 4 according accord VERB ejde-39 62 5 to to ADP ejde-39 62 6 their their PRON ejde-39 62 7 finite finite ADJ ejde-39 62 8 multiplicity multiplicity NOUN ejde-39 62 9 . . PUNCT ejde-39 63 1 now now ADV ejde-39 63 2 , , PUNCT ejde-39 63 3 with with ADP ejde-39 63 4 the the DET ejde-39 63 5 convention convention NOUN ejde-39 63 6 that that SCONJ ejde-39 63 7 λ0 λ0 NOUN ejde-39 63 8 = = SYM ejde-39 63 9 −∞ −∞ NOUN ejde-39 63 10 , , PUNCT ejde-39 63 11 our our PRON ejde-39 63 12 main main ADJ ejde-39 63 13 result result NOUN ejde-39 63 14 is be AUX ejde-39 63 15 as as SCONJ ejde-39 63 16 follows follow NOUN ejde-39 63 17 . . PUNCT ejde-39 64 1 theorem theorem ADJ ejde-39 64 2 1.1 1.1 NUM ejde-39 64 3 . . PUNCT ejde-39 65 1 assume assume VERB ejde-39 65 2 that that SCONJ ejde-39 65 3 1 1 NUM ejde-39 65 4 < < X ejde-39 65 5 q q X ejde-39 65 6 < < X ejde-39 65 7 2 2 NUM ejde-39 65 8 < < X ejde-39 65 9 γ γ X ejde-39 65 10 < < X ejde-39 65 11 2∗ 2∗ PROPN ejde-39 65 12 , , PUNCT ejde-39 65 13 λ λ PROPN ejde-39 65 14 ∈ ∈ PROPN ejde-39 65 15 ( ( PUNCT ejde-39 65 16 λk λk X ejde-39 65 17 , , PUNCT ejde-39 65 18 λk+1 λk+1 NUM ejde-39 65 19 ) ) PUNCT ejde-39 65 20 , , PUNCT ejde-39 65 21 k k X ejde-39 66 1 = = PUNCT ejde-39 66 2 0 0 PROPN ejde-39 66 3 , , PUNCT ejde-39 66 4 . . PUNCT ejde-39 66 5 . . PUNCT ejde-39 67 1 . . PUNCT ejde-39 68 1 , , PUNCT ejde-39 68 2 and and CCONJ ejde-39 68 3 ( ( PUNCT ejde-39 68 4 a1)-(a2 a1)-(a2 PROPN ejde-39 68 5 ) ) PUNCT ejde-39 68 6 hold hold NOUN ejde-39 68 7 . . PUNCT ejde-39 69 1 then then ADV ejde-39 69 2 there there PRON ejde-39 69 3 exists exist VERB ejde-39 69 4 ekλ ekλ VERB ejde-39 69 5 > > X ejde-39 69 6 0 0 NUM ejde-39 70 1 such such ADJ ejde-39 70 2 that that PRON ejde-39 70 3 for for ADP ejde-39 70 4 any any DET ejde-39 70 5 given give VERB ejde-39 70 6 e e NOUN ejde-39 70 7 ∈ ∈ PROPN ejde-39 70 8 ( ( PUNCT ejde-39 70 9 0 0 NUM ejde-39 70 10 , , PUNCT ejde-39 70 11 ekλ ekλ ADJ ejde-39 70 12 ) ) PUNCT ejde-39 70 13 corresponds correspond VERB ejde-39 70 14 µkλ(e µkλ(e NOUN ejde-39 70 15 ) ) PUNCT ejde-39 70 16 ∈ ∈ PROPN ejde-39 70 17 ( ( PUNCT ejde-39 70 18 0,+∞ 0,+∞ NUM ejde-39 70 19 ) ) PUNCT ejde-39 70 20 such such ADJ ejde-39 70 21 that that SCONJ ejde-39 70 22 ( ( PUNCT ejde-39 70 23 1.1 1.1 NUM ejde-39 70 24 ) ) PUNCT ejde-39 70 25 with with ADP ejde-39 70 26 µ µ NOUN ejde-39 70 27 = = SYM ejde-39 70 28 µkλ(e µkλ(e PROPN ejde-39 70 29 ) ) PUNCT ejde-39 70 30 possesses possess VERB ejde-39 70 31 a a DET ejde-39 70 32 non non ADJ ejde-39 70 33 - - ADJ ejde-39 70 34 zero zero ADJ ejde-39 70 35 weak weak ADJ ejde-39 70 36 solution solution NOUN ejde-39 70 37 uµkλ(e uµkλ(e NOUN ejde-39 70 38 ) ) PUNCT ejde-39 70 39 ∈ ∈ NOUN ejde-39 70 40 c1,α(ω c1,α(ω NOUN ejde-39 70 41 ) ) PUNCT ejde-39 70 42 , , PUNCT ejde-39 70 43 α α PROPN ejde-39 70 44 ∈ ∈ PROPN ejde-39 70 45 ( ( PUNCT ejde-39 70 46 0 0 NUM ejde-39 70 47 , , PUNCT ejde-39 70 48 1 1 NUM ejde-39 70 49 ) ) PUNCT ejde-39 70 50 with with ADP ejde-39 70 51 energy energy NOUN ejde-39 70 52 value value NOUN ejde-39 70 53 e e NOUN ejde-39 70 54 , , PUNCT ejde-39 70 55 i.e. i.e. X ejde-39 70 56 , , PUNCT ejde-39 70 57 deµkλ(e)(uµkλ(e deµkλ(e)(uµkλ(e NUM ejde-39 70 58 ) ) PUNCT ejde-39 70 59 ) ) PUNCT ejde-39 71 1 = = SYM ejde-39 71 2 0 0 NUM ejde-39 71 3 , , PUNCT ejde-39 71 4 eµkλ(e)(uµkλ(e eµkλ(e)(uµkλ(e X ejde-39 71 5 ) ) PUNCT ejde-39 71 6 ) ) PUNCT ejde-39 72 1 = = SYM ejde-39 72 2 e. e. PROPN ejde-39 72 3 furthermore furthermore ADV ejde-39 72 4 , , PUNCT ejde-39 72 5 ( ( PUNCT ejde-39 72 6 i i NOUN ejde-39 72 7 ) ) PUNCT ejde-39 72 8 µkλ µkλ PROPN ejde-39 72 9 ( ( PUNCT ejde-39 72 10 · · PUNCT ejde-39 72 11 ) ) PUNCT ejde-39 72 12 is be AUX ejde-39 72 13 a a DET ejde-39 72 14 non non ADJ ejde-39 72 15 - - ADJ ejde-39 72 16 increasing increasing ADJ ejde-39 72 17 function function NOUN ejde-39 72 18 in in ADP ejde-39 72 19 ( ( PUNCT ejde-39 72 20 0 0 NUM ejde-39 72 21 , , PUNCT ejde-39 72 22 ekλ ekλ ADJ ejde-39 72 23 ) ) PUNCT ejde-39 72 24 ; ; PUNCT ejde-39 72 25 ( ( PUNCT ejde-39 72 26 ii ii NOUN ejde-39 72 27 ) ) PUNCT ejde-39 72 28 if if SCONJ ejde-39 72 29 λ λ PROPN ejde-39 72 30 < < X ejde-39 72 31 λ1 λ1 PROPN ejde-39 72 32 , , PUNCT ejde-39 72 33 then then ADV ejde-39 72 34 there there PRON ejde-39 72 35 exists exist VERB ejde-39 72 36 lime→0 lime→0 PROPN ejde-39 72 37 µ µ X ejde-39 72 38 0 0 NUM ejde-39 72 39 λ(e λ(e ADJ ejde-39 72 40 ) ) PUNCT ejde-39 72 41 = = SYM ejde-39 73 1 µ̄λ(0 µ̄λ(0 NOUN ejde-39 73 2 ) ) PUNCT ejde-39 73 3 ∈ ∈ PROPN ejde-39 73 4 ( ( PUNCT ejde-39 73 5 0,+∞ 0,+∞ NUM ejde-39 73 6 ) ) PUNCT ejde-39 73 7 such such ADJ ejde-39 73 8 that that SCONJ ejde-39 73 9 ( ( PUNCT ejde-39 73 10 1.1 1.1 NUM ejde-39 73 11 ) ) PUNCT ejde-39 73 12 possesses possess VERB ejde-39 73 13 a a DET ejde-39 73 14 non non ADJ ejde-39 73 15 - - ADJ ejde-39 73 16 zero zero ADJ ejde-39 73 17 weak weak ADJ ejde-39 73 18 solution solution NOUN ejde-39 73 19 uµ̄λ(0 uµ̄λ(0 NOUN ejde-39 73 20 ) ) PUNCT ejde-39 73 21 ∈ ∈ PROPN ejde-39 73 22 c1,α(ω c1,α(ω NOUN ejde-39 73 23 ) ) PUNCT ejde-39 73 24 , , PUNCT ejde-39 73 25 α α PROPN ejde-39 73 26 ∈ ∈ PROPN ejde-39 73 27 ( ( PUNCT ejde-39 73 28 0 0 NUM ejde-39 73 29 , , PUNCT ejde-39 73 30 1 1 NUM ejde-39 73 31 ) ) PUNCT ejde-39 73 32 with with ADP ejde-39 73 33 zero zero NUM ejde-39 73 34 energy energy NOUN ejde-39 73 35 value value NOUN ejde-39 73 36 e e NOUN ejde-39 73 37 = = SYM ejde-39 73 38 0 0 NUM ejde-39 73 39 and and CCONJ ejde-39 73 40 µ µ X ejde-39 73 41 = = SYM ejde-39 73 42 µ̄λ(0 µ̄λ(0 NOUN ejde-39 73 43 ) ) PUNCT ejde-39 73 44 . . PUNCT ejde-39 74 1 to to PART ejde-39 74 2 find find VERB ejde-39 74 3 solutions solution NOUN ejde-39 74 4 of of ADP ejde-39 74 5 ( ( PUNCT ejde-39 74 6 1.1 1.1 NUM ejde-39 74 7 ) ) PUNCT ejde-39 74 8 , , PUNCT ejde-39 74 9 we we PRON ejde-39 74 10 apply apply VERB ejde-39 74 11 the the DET ejde-39 74 12 benci benci PROPN ejde-39 74 13 - - PUNCT ejde-39 74 14 rabinowitz rabinowitz PROPN ejde-39 74 15 linking linking NOUN ejde-39 74 16 theorem theorem NOUN ejde-39 74 17 [ [ X ejde-39 74 18 5 5 NUM ejde-39 74 19 ] ] PUNCT ejde-39 74 20 to to ADP ejde-39 74 21 the the DET ejde-39 74 22 energy energy NOUN ejde-39 74 23 level level NOUN ejde-39 74 24 nonlinear nonlinear PROPN ejde-39 74 25 rayleigh rayleigh PROPN ejde-39 74 26 quotient quotient NOUN ejde-39 74 27 reλ reλ NOUN ejde-39 74 28 ( ( PUNCT ejde-39 74 29 u u NOUN ejde-39 74 30 ) ) PUNCT ejde-39 74 31 . . PUNCT ejde-39 75 1 note note VERB ejde-39 75 2 that that SCONJ ejde-39 75 3 reλ reλ NOUN ejde-39 75 4 ∈ ∈ PROPN ejde-39 75 5 c1(ẘ c1(ẘ NOUN ejde-39 75 6 1 1 NUM ejde-39 75 7 2 2 NUM ejde-39 75 8 ( ( PUNCT ejde-39 75 9 ω ω NOUN ejde-39 75 10 ) ) PUNCT ejde-39 75 11 \ \ NOUN ejde-39 75 12 { { PUNCT ejde-39 75 13 0},r 0},r NOUN ejde-39 75 14 ) ) PUNCT ejde-39 75 15 , , PUNCT ejde-39 75 16 while while SCONJ ejde-39 75 17 for for ADP ejde-39 75 18 the the DET ejde-39 75 19 application application NOUN ejde-39 75 20 of of ADP ejde-39 75 21 the the DET ejde-39 75 22 linking linking NOUN ejde-39 75 23 theorem theorem NOUN ejde-39 75 24 , , PUNCT ejde-39 75 25 in in ADP ejde-39 75 26 general general ADJ ejde-39 75 27 , , PUNCT ejde-39 75 28 it it PRON ejde-39 75 29 is be AUX ejde-39 75 30 required require VERB ejde-39 75 31 that that SCONJ ejde-39 75 32 the the DET ejde-39 75 33 functional functional ADJ ejde-39 75 34 belongs belong VERB ejde-39 75 35 to to ADP ejde-39 75 36 c1(ẘ c1(ẘ NOUN ejde-39 75 37 1 1 NUM ejde-39 75 38 2 2 NUM ejde-39 75 39 ( ( PUNCT ejde-39 75 40 ω),r ω),r NOUN ejde-39 75 41 ) ) PUNCT ejde-39 75 42 . . PUNCT ejde-39 76 1 below below ADP ejde-39 76 2 we we PRON ejde-39 76 3 overcome overcome VERB ejde-39 76 4 this this DET ejde-39 76 5 difficulty difficulty NOUN ejde-39 76 6 by by ADP ejde-39 76 7 using use VERB ejde-39 76 8 an an DET ejde-39 76 9 appropriate appropriate ADJ ejde-39 76 10 truncation truncation NOUN ejde-39 76 11 function function NOUN ejde-39 76 12 for for ADP ejde-39 76 13 reλ reλ NOUN ejde-39 76 14 which which PRON ejde-39 76 15 can can AUX ejde-39 76 16 be be AUX ejde-39 76 17 properly properly ADV ejde-39 76 18 introduced introduce VERB ejde-39 76 19 in in ADP ejde-39 76 20 the the DET ejde-39 76 21 case case NOUN ejde-39 76 22 e e X ejde-39 76 23 > > X ejde-39 76 24 0 0 X ejde-39 76 25 . . PUNCT ejde-39 77 1 in in ADP ejde-39 77 2 the the DET ejde-39 77 3 zero zero NUM ejde-39 77 4 - - PUNCT ejde-39 77 5 energy energy NOUN ejde-39 77 6 case case NOUN ejde-39 77 7 e e X ejde-39 77 8 = = SYM ejde-39 77 9 0 0 PROPN ejde-39 77 10 , , PUNCT ejde-39 77 11 the the DET ejde-39 77 12 solution solution NOUN ejde-39 77 13 is be AUX ejde-39 77 14 obtained obtain VERB ejde-39 77 15 by by ADP ejde-39 77 16 passing pass VERB ejde-39 77 17 to to ADP ejde-39 77 18 the the DET ejde-39 77 19 limit limit NOUN ejde-39 77 20 e e X ejde-39 77 21 → → X ejde-39 77 22 0 0 X ejde-39 77 23 . . PUNCT ejde-39 77 24 remark remark PROPN ejde-39 77 25 1.2 1.2 NUM ejde-39 77 26 . . PUNCT ejde-39 78 1 condition condition NOUN ejde-39 78 2 ( ( PUNCT ejde-39 78 3 a2 a2 PROPN ejde-39 78 4 ) ) PUNCT ejde-39 78 5 is be AUX ejde-39 78 6 the the DET ejde-39 78 7 well well ADV ejde-39 78 8 - - PUNCT ejde-39 78 9 known know VERB ejde-39 78 10 ambrosetti ambrosetti NOUN ejde-39 78 11 - - PUNCT ejde-39 78 12 rabinovich rabinovich NOUN ejde-39 78 13 ( ( PUNCT ejde-39 78 14 ar ar NOUN ejde-39 78 15 ) ) PUNCT ejde-39 78 16 condition condition NOUN ejde-39 79 1 [ [ X ejde-39 79 2 1 1 NUM ejde-39 79 3 , , PUNCT ejde-39 79 4 15 15 NUM ejde-39 79 5 ] ] PUNCT ejde-39 80 1 and and CCONJ ejde-39 80 2 implies imply VERB ejde-39 80 3 the the DET ejde-39 80 4 superquadratic superquadratic ADJ ejde-39 80 5 behavior behavior NOUN ejde-39 80 6 of of ADP ejde-39 80 7 g g PROPN ejde-39 80 8 ( ( PUNCT ejde-39 80 9 · · PUNCT ejde-39 80 10 , , PUNCT ejde-39 80 11 s s PART ejde-39 80 12 ) ) PUNCT ejde-39 80 13 , , PUNCT ejde-39 80 14 g(x g(x PROPN ejde-39 80 15 , , PUNCT ejde-39 80 16 s s NOUN ejde-39 80 17 ) ) PUNCT ejde-39 80 18 ≥ ≥ NOUN ejde-39 80 19 c|s|α c|s|α VERB ejde-39 80 20 for for ADP ejde-39 80 21 some some DET ejde-39 80 22 c c PROPN ejde-39 80 23 > > X ejde-39 80 24 0 0 PUNCT ejde-39 80 25 and and CCONJ ejde-39 80 26 |s| |s| NOUN ejde-39 80 27 large large ADJ ejde-39 80 28 . . PUNCT ejde-39 81 1 however however ADV ejde-39 81 2 , , PUNCT ejde-39 81 3 the the DET ejde-39 81 4 complete complete ADJ ejde-39 81 5 nonlinearity nonlinearity NOUN ejde-39 81 6 µ|u|q−1u+ µ|u|q−1u+ NOUN ejde-39 81 7 g(x g(x NOUN ejde-39 81 8 , , PUNCT ejde-39 81 9 u u NOUN ejde-39 81 10 ) ) PUNCT ejde-39 81 11 of of ADP ejde-39 81 12 equation equation NOUN ejde-39 81 13 ( ( PUNCT ejde-39 81 14 1.1 1.1 NUM ejde-39 81 15 ) ) PUNCT ejde-39 81 16 does do AUX ejde-39 81 17 not not PART ejde-39 81 18 satisfy satisfy VERB ejde-39 81 19 the the DET ejde-39 81 20 ( ( PUNCT ejde-39 81 21 ar ar NOUN ejde-39 81 22 ) ) PUNCT ejde-39 81 23 condition condition NOUN ejde-39 81 24 . . PUNCT ejde-39 82 1 remark remark NOUN ejde-39 82 2 1.3 1.3 NUM ejde-39 82 3 . . PUNCT ejde-39 83 1 the the DET ejde-39 83 2 zero zero NUM ejde-39 83 3 - - PUNCT ejde-39 83 4 energy energy NOUN ejde-39 83 5 case case NOUN ejde-39 83 6 e e NOUN ejde-39 83 7 = = SYM ejde-39 83 8 0 0 NUM ejde-39 83 9 is be AUX ejde-39 83 10 particularly particularly ADV ejde-39 83 11 interesting interesting ADJ ejde-39 83 12 , , PUNCT ejde-39 83 13 since since SCONJ ejde-39 83 14 the the DET ejde-39 83 15 value value NOUN ejde-39 83 16 µ̄λ(0 µ̄λ(0 NOUN ejde-39 83 17 ) ) PUNCT ejde-39 83 18 in in ADP ejde-39 83 19 this this DET ejde-39 83 20 case case NOUN ejde-39 83 21 resembles resemble VERB ejde-39 83 22 linear linear PROPN ejde-39 83 23 eigenvalue eigenvalue PROPN ejde-39 83 24 . . PUNCT ejde-39 84 1 indeed indeed ADV ejde-39 84 2 , , PUNCT ejde-39 84 3 for for ADP ejde-39 84 4 linear linear NOUN ejde-39 84 5 problems problem NOUN ejde-39 84 6 such such ADJ ejde-39 84 7 as as ADP ejde-39 84 8 lu lu NOUN ejde-39 84 9 = = SYM ejde-39 84 10 λu λu PROPN ejde-39 84 11 , , PUNCT ejde-39 84 12 where where SCONJ ejde-39 84 13 l l NOUN ejde-39 84 14 is be AUX ejde-39 84 15 a a DET ejde-39 84 16 self self NOUN ejde-39 84 17 - - PUNCT ejde-39 84 18 adjoint adjoint NOUN ejde-39 84 19 linear linear ADJ ejde-39 84 20 operator operator NOUN ejde-39 84 21 on on ADP ejde-39 84 22 a a DET ejde-39 84 23 hilbert hilbert NOUN ejde-39 84 24 space space NOUN ejde-39 84 25 h h NOUN ejde-39 84 26 , , PUNCT ejde-39 84 27 4 4 NUM ejde-39 84 28 y. y. NOUN ejde-39 84 29 il’yasov il’yasov PROPN ejde-39 84 30 , , PUNCT ejde-39 84 31 e. e. PROPN ejde-39 84 32 d. d. PROPN ejde-39 84 33 silva silva PROPN ejde-39 84 34 , , PUNCT ejde-39 84 35 m. m. PROPN ejde-39 84 36 l. l. PROPN ejde-39 84 37 silva silva PROPN ejde-39 84 38 ejde-2023/23 ejde-2023/23 PROPN ejde-39 84 39 any any DET ejde-39 84 40 isolated isolated ADJ ejde-39 84 41 eigenvalue eigenvalue PROPN ejde-39 84 42 λn λn NOUN ejde-39 84 43 , , PUNCT ejde-39 84 44 n n NOUN ejde-39 84 45 = = SYM ejde-39 84 46 1 1 NUM ejde-39 84 47 , , PUNCT ejde-39 84 48 2 2 NUM ejde-39 84 49 , , PUNCT ejde-39 84 50 . . PUNCT ejde-39 84 51 . . PUNCT ejde-39 84 52 . . PUNCT ejde-39 85 1 , , PUNCT ejde-39 85 2 corresponds correspond VERB ejde-39 85 3 to to ADP ejde-39 85 4 an an DET ejde-39 85 5 eigenfunction eigenfunction NOUN ejde-39 85 6 φn φn ADP ejde-39 85 7 of of ADP ejde-39 85 8 the the DET ejde-39 85 9 zero zero NUM ejde-39 85 10 - - PUNCT ejde-39 85 11 energy energy NOUN ejde-39 85 12 level level NOUN ejde-39 85 13 , , PUNCT ejde-39 85 14 i. i. PROPN ejde-39 85 15 e. e. PROPN ejde-39 85 16 , , PUNCT ejde-39 85 17 e e PROPN ejde-39 85 18 = = NOUN ejde-39 85 19 1 1 NUM ejde-39 85 20 2 2 NUM ejde-39 85 21 〈 〈 PROPN ejde-39 85 22 φn φn NOUN ejde-39 85 23 , , PUNCT ejde-39 85 24 lφn lφn PROPN ejde-39 85 25 〉 〉 NOUN ejde-39 86 1 − − NOUN ejde-39 87 1 λn λn NOUN ejde-39 87 2 1 1 NUM ejde-39 87 3 2 2 NUM ejde-39 87 4 〈 〈 PROPN ejde-39 87 5 φn φn NOUN ejde-39 87 6 , , PUNCT ejde-39 87 7 φn φn PROPN ejde-39 87 8 〉 〉 NOUN ejde-39 87 9 = = SYM ejde-39 87 10 0 0 X ejde-39 87 11 . . PUNCT ejde-39 88 1 we we PRON ejde-39 88 2 shall shall AUX ejde-39 88 3 use use VERB ejde-39 88 4 the the DET ejde-39 88 5 following following ADJ ejde-39 88 6 notation notation NOUN ejde-39 88 7 : : PUNCT ejde-39 88 8 • • NOUN ejde-39 88 9 w w NOUN ejde-39 88 10 : : PUNCT ejde-39 88 11 = = PUNCT ejde-39 88 12 ẘ ẘ VERB ejde-39 88 13 1 1 NUM ejde-39 88 14 2 2 NUM ejde-39 88 15 ( ( PUNCT ejde-39 88 16 ω ω NOUN ejde-39 88 17 ) ) PUNCT ejde-39 88 18 denotes denote VERB ejde-39 88 19 the the DET ejde-39 88 20 standard standard ADJ ejde-39 88 21 sobolev sobolev ADJ ejde-39 88 22 space space NOUN ejde-39 88 23 with with ADP ejde-39 88 24 the the DET ejde-39 88 25 norm norm NOUN ejde-39 88 26 ‖u‖w ‖u‖w NOUN ejde-39 88 27 = = SYM ejde-39 88 28 ( ( PUNCT ejde-39 88 29 ∫ ∫ PROPN ejde-39 88 30 ω ω PROPN ejde-39 88 31 |∇u|2 |∇u|2 NOUN ejde-39 88 32 dx)1/2 dx)1/2 PROPN ejde-39 88 33 ; ; PUNCT ejde-39 88 34 • • ADP ejde-39 88 35 |u|lr |u|lr NUM ejde-39 88 36 : : PUNCT ejde-39 88 37 = = SYM ejde-39 88 38 ( ( PUNCT ejde-39 88 39 ∫ ∫ PROPN ejde-39 88 40 ω ω PROPN ejde-39 89 1 |u|r |u|r PROPN ejde-39 89 2 dx)1 dx)1 PROPN ejde-39 89 3 / / SYM ejde-39 89 4 r r NOUN ejde-39 89 5 , , PUNCT ejde-39 89 6 1 1 NUM ejde-39 89 7 ≤ ≤ NOUN ejde-39 89 8 r r NOUN ejde-39 89 9 < < X ejde-39 89 10 + + NOUN ejde-39 89 11 ∞ ∞ PROPN ejde-39 89 12 denotes denote VERB ejde-39 89 13 the the DET ejde-39 89 14 norm norm NOUN ejde-39 89 15 on on ADP ejde-39 89 16 the the DET ejde-39 89 17 lebesgue lebesgue NOUN ejde-39 89 18 space space NOUN ejde-39 89 19 lr lr X ejde-39 89 20 : : PUNCT ejde-39 89 21 = = NOUN ejde-39 89 22 lr(ω lr(ω X ejde-39 89 23 ) ) PUNCT ejde-39 89 24 , , PUNCT ejde-39 89 25 ( ( PUNCT ejde-39 89 26 · · PUNCT ejde-39 89 27 , , PUNCT ejde-39 89 28 · · PUNCT ejde-39 89 29 ) ) PUNCT ejde-39 89 30 denotes denote VERB ejde-39 89 31 the the DET ejde-39 89 32 scalar scalar ADJ ejde-39 89 33 product product NOUN ejde-39 89 34 in in ADP ejde-39 89 35 l2 l2 NOUN ejde-39 89 36 ; ; PUNCT ejde-39 89 37 • • NUM ejde-39 89 38 sp sp NOUN ejde-39 89 39 is be AUX ejde-39 89 40 the the DET ejde-39 89 41 best good ADJ ejde-39 89 42 sobolev sobolev NOUN ejde-39 89 43 constant constant ADJ ejde-39 89 44 for for ADP ejde-39 89 45 the the DET ejde-39 89 46 embedding embed VERB ejde-39 89 47 w w PROPN ejde-39 89 48 1,2 1,2 NUM ejde-39 89 49 0 0 NUM ejde-39 89 50 ( ( PUNCT ejde-39 89 51 ω ω NOUN ejde-39 89 52 ) ) PUNCT ejde-39 89 53 ⊂ ⊂ PROPN ejde-39 89 54 lp(ω lp(ω PROPN ejde-39 89 55 ) ) PUNCT ejde-39 89 56 , , PUNCT ejde-39 89 57 1 1 NUM ejde-39 89 58 ≤ ≤ NOUN ejde-39 89 59 p p NOUN ejde-39 89 60 ≤ ≤ NUM ejde-39 89 61 2∗ 2∗ NUM ejde-39 89 62 ; ; PUNCT ejde-39 89 63 • • NUM ejde-39 89 64 ‖ ‖ PROPN ejde-39 89 65 · · PUNCT ejde-39 89 66 ‖∗ ‖∗ PROPN ejde-39 89 67 denotes denote VERB ejde-39 89 68 the the DET ejde-39 89 69 norm norm NOUN ejde-39 89 70 in in ADP ejde-39 89 71 the the DET ejde-39 89 72 dual dual ADJ ejde-39 89 73 space space NOUN ejde-39 90 1 w w NOUN ejde-39 90 2 ∗ ∗ NOUN ejde-39 90 3 ; ; PUNCT ejde-39 90 4 • • NUM ejde-39 90 5 d(a d(a PROPN ejde-39 90 6 , , PUNCT ejde-39 90 7 b b NOUN ejde-39 90 8 ) ) PUNCT ejde-39 90 9 : : PUNCT ejde-39 91 1 = = SYM ejde-39 91 2 min{‖u min{‖u PROPN ejde-39 91 3 − − PROPN ejde-39 91 4 v‖w v‖w NOUN ejde-39 91 5 : : PUNCT ejde-39 91 6 u u PROPN ejde-39 91 7 ∈ ∈ PROPN ejde-39 91 8 a a PRON ejde-39 91 9 , , PUNCT ejde-39 91 10 v v PROPN ejde-39 91 11 ∈ ∈ PROPN ejde-39 91 12 b b NOUN ejde-39 91 13 } } PUNCT ejde-39 91 14 denotes denote VERB ejde-39 91 15 the the DET ejde-39 91 16 distance distance NOUN ejde-39 91 17 between between ADP ejde-39 91 18 sets set NOUN ejde-39 91 19 a a DET ejde-39 91 20 , , PUNCT ejde-39 91 21 b b PROPN ejde-39 91 22 ⊂w ⊂w PROPN ejde-39 91 23 . . PUNCT ejde-39 92 1 this this DET ejde-39 92 2 work work NOUN ejde-39 92 3 is be AUX ejde-39 92 4 organized organize VERB ejde-39 92 5 as as SCONJ ejde-39 92 6 follows follow VERB ejde-39 92 7 . . PUNCT ejde-39 93 1 in in ADP ejde-39 93 2 section section NOUN ejde-39 93 3 2 2 NUM ejde-39 93 4 , , PUNCT ejde-39 93 5 we we PRON ejde-39 93 6 introduce introduce VERB ejde-39 93 7 the the DET ejde-39 93 8 nonlinear nonlinear ADJ ejde-39 93 9 rayleigh rayleigh PROPN ejde-39 93 10 quotient quotient NOUN ejde-39 93 11 together together ADV ejde-39 93 12 with with ADP ejde-39 93 13 its its PRON ejde-39 93 14 appropriate appropriate ADJ ejde-39 93 15 truncation truncation NOUN ejde-39 93 16 function function NOUN ejde-39 93 17 . . PUNCT ejde-39 94 1 in in ADP ejde-39 94 2 section section NOUN ejde-39 94 3 3 3 NUM ejde-39 94 4 , , PUNCT ejde-39 94 5 we we PRON ejde-39 94 6 derive derive VERB ejde-39 94 7 some some DET ejde-39 94 8 properties property NOUN ejde-39 94 9 of of ADP ejde-39 94 10 the the DET ejde-39 94 11 nonlinear nonlinear PROPN ejde-39 94 12 rayleigh rayleigh PROPN ejde-39 94 13 quotient quotient NOUN ejde-39 94 14 reλ reλ NOUN ejde-39 94 15 and and CCONJ ejde-39 94 16 prove prove VERB ejde-39 94 17 that that SCONJ ejde-39 94 18 the the DET ejde-39 94 19 cerami cerami PROPN ejde-39 94 20 condition condition NOUN ejde-39 94 21 for for ADP ejde-39 94 22 reλ reλ NOUN ejde-39 94 23 is be AUX ejde-39 94 24 satisfied satisfied ADJ ejde-39 94 25 . . PUNCT ejde-39 95 1 in in ADP ejde-39 95 2 section section NOUN ejde-39 95 3 4 4 NUM ejde-39 95 4 , , PUNCT ejde-39 95 5 we we PRON ejde-39 95 6 prove prove VERB ejde-39 95 7 our our PRON ejde-39 95 8 main main ADJ ejde-39 95 9 result result NOUN ejde-39 95 10 . . PUNCT ejde-39 96 1 conclusions conclusion NOUN ejde-39 96 2 are be AUX ejde-39 96 3 drawn draw VERB ejde-39 96 4 in in ADP ejde-39 96 5 section section NOUN ejde-39 96 6 5 5 NUM ejde-39 96 7 . . PUNCT ejde-39 97 1 in in ADP ejde-39 97 2 the the DET ejde-39 97 3 appendix appendix NOUN ejde-39 97 4 , , PUNCT ejde-39 97 5 we we PRON ejde-39 97 6 state state VERB ejde-39 97 7 the the DET ejde-39 97 8 bencirabinowitz bencirabinowitz NOUN ejde-39 97 9 linking link VERB ejde-39 97 10 theorem theorem NOUN ejde-39 97 11 and and CCONJ ejde-39 97 12 corresponding corresponding ADJ ejde-39 97 13 definitions definition NOUN ejde-39 97 14 . . PUNCT ejde-39 98 1 2 2 X ejde-39 98 2 . . X ejde-39 98 3 preliminaries preliminary NOUN ejde-39 98 4 let let VERB ejde-39 98 5 ( ( PUNCT ejde-39 98 6 ek ek NOUN ejde-39 98 7 ) ) PUNCT ejde-39 98 8 ⊂ ⊂ PROPN ejde-39 99 1 w w VERB ejde-39 99 2 be be AUX ejde-39 99 3 the the DET ejde-39 99 4 orthogonal orthogonal ADJ ejde-39 99 5 basis basis NOUN ejde-39 99 6 in in ADP ejde-39 99 7 l2 l2 NOUN ejde-39 99 8 of of ADP ejde-39 99 9 the the DET ejde-39 99 10 eigenfunctions eigenfunction NOUN ejde-39 99 11 of of ADP ejde-39 99 12 ( ( PUNCT ejde-39 99 13 −∆ −∆ NOUN ejde-39 99 14 ) ) PUNCT ejde-39 99 15 with with ADP ejde-39 99 16 zero zero NUM ejde-39 99 17 dirichlet dirichlet NOUN ejde-39 99 18 conditions condition NOUN ejde-39 99 19 satisfying satisfy VERB ejde-39 99 20 ‖ek‖2w ‖ek‖2w NOUN ejde-39 99 21 = = PUNCT ejde-39 99 22 λk λk PROPN ejde-39 99 23 and and CCONJ ejde-39 99 24 |ek|2l2 |ek|2l2 PROPN ejde-39 99 25 = = SYM ejde-39 99 26 1 1 NUM ejde-39 99 27 , , PUNCT ejde-39 99 28 for for SCONJ ejde-39 99 29 each each DET ejde-39 99 30 k k PROPN ejde-39 99 31 ∈ ∈ PROPN ejde-39 99 32 n. n. NOUN ejde-39 99 33 let let VERB ejde-39 99 34 λ λ X ejde-39 99 35 ∈ ∈ PROPN ejde-39 99 36 ( ( PUNCT ejde-39 99 37 λk λk X ejde-39 99 38 , , PUNCT ejde-39 99 39 λk+1 λk+1 X ejde-39 99 40 ) ) PUNCT ejde-39 99 41 be be AUX ejde-39 99 42 fixed fix VERB ejde-39 99 43 for for ADP ejde-39 99 44 some some DET ejde-39 99 45 k k PROPN ejde-39 99 46 ∈ ∈ PROPN ejde-39 99 47 n. n. NOUN ejde-39 99 48 we we PRON ejde-39 99 49 write write VERB ejde-39 99 50 w w PROPN ejde-39 99 51 = = PUNCT ejde-39 99 52 w+ w+ PUNCT ejde-39 99 53 ⊕w− ⊕w− NOUN ejde-39 99 54 , , PUNCT ejde-39 99 55 where where SCONJ ejde-39 99 56 w− w− NOUN ejde-39 99 57 = = SYM ejde-39 99 58 span{e1 span{e1 NOUN ejde-39 99 59 , , PUNCT ejde-39 99 60 e2 e2 PROPN ejde-39 99 61 , , PUNCT ejde-39 99 62 . . PUNCT ejde-39 99 63 . . PUNCT ejde-39 99 64 . . PUNCT ejde-39 100 1 , , PUNCT ejde-39 100 2 ek ek PROPN ejde-39 100 3 } } PUNCT ejde-39 100 4 , , PUNCT ejde-39 100 5 w+ w+ NOUN ejde-39 100 6 = = SYM ejde-39 100 7 span{ek+1 span{ek+1 NOUN ejde-39 100 8 , , PUNCT ejde-39 100 9 ek+2 ek+2 NOUN ejde-39 100 10 , , PUNCT ejde-39 100 11 . . PUNCT ejde-39 100 12 . . PUNCT ejde-39 101 1 . . PUNCT ejde-39 101 2 } } PUNCT ejde-39 101 3 . . PUNCT ejde-39 102 1 then then ADV ejde-39 102 2 one one PRON ejde-39 102 3 can can AUX ejde-39 102 4 introduce introduce VERB ejde-39 102 5 the the DET ejde-39 102 6 following follow VERB ejde-39 102 7 equivalent equivalent ADJ ejde-39 102 8 norm norm NOUN ejde-39 102 9 ‖ ‖ PROPN ejde-39 102 10 · · PUNCT ejde-39 102 11 ‖1 ‖1 NOUN ejde-39 102 12 for for ADP ejde-39 102 13 ‖ ‖ PROPN ejde-39 102 14 · · PUNCT ejde-39 102 15 ‖w ‖w PROPN ejde-39 102 16 in in ADP ejde-39 102 17 w w NOUN ejde-39 102 18 ‖u‖21 ‖u‖21 NOUN ejde-39 102 19 = = SYM ejde-39 102 20 ∞∑ ∞∑ NUM ejde-39 102 21 i i PRON ejde-39 102 22 = = NOUN ejde-39 102 23 k+1 k+1 X ejde-39 102 24 ( ( PUNCT ejde-39 102 25 λi λi ADP ejde-39 102 26 − − PROPN ejde-39 102 27 λ)u2 λ)u2 PROPN ejde-39 102 28 i i PRON ejde-39 102 29 + + CCONJ ejde-39 102 30 k∑ k∑ ADJ ejde-39 102 31 i=1 i=1 X ejde-39 103 1 ( ( PUNCT ejde-39 103 2 λ− λ− PROPN ejde-39 103 3 λi)u2 λi)u2 PROPN ejde-39 103 4 1 1 NUM ejde-39 103 5 : : PUNCT ejde-39 103 6 = = PUNCT ejde-39 103 7 ‖u+‖21 ‖u+‖21 NOUN ejde-39 103 8 + + CCONJ ejde-39 103 9 ‖u−‖21 ‖u−‖21 NOUN ejde-39 103 10 , , PUNCT ejde-39 103 11 where where SCONJ ejde-39 103 12 ui ui PROPN ejde-39 103 13 = = PUNCT ejde-39 103 14 ( ( PUNCT ejde-39 103 15 u u NOUN ejde-39 103 16 , , PUNCT ejde-39 103 17 ei ei PROPN ejde-39 103 18 ) ) PUNCT ejde-39 103 19 , , PUNCT ejde-39 103 20 i i PRON ejde-39 103 21 = = NOUN ejde-39 103 22 1 1 NUM ejde-39 103 23 , , PUNCT ejde-39 103 24 . . PUNCT ejde-39 103 25 . . PUNCT ejde-39 104 1 .. .. PUNCT ejde-39 105 1 then then ADV ejde-39 105 2 u u X ejde-39 105 3 = = PUNCT ejde-39 105 4 ( ( PUNCT ejde-39 105 5 u+ u+ X ejde-39 105 6 + + NUM ejde-39 105 7 u− u− ADJ ejde-39 105 8 ) ) PUNCT ejde-39 105 9 ∈ ∈ PROPN ejde-39 105 10 w w PROPN ejde-39 105 11 , , PUNCT ejde-39 105 12 u± u± PROPN ejde-39 105 13 ∈ ∈ PROPN ejde-39 105 14 w± w± NOUN ejde-39 105 15 , , PUNCT ejde-39 105 16 and and CCONJ ejde-39 105 17 c0‖u‖21 c0‖u‖21 NUM ejde-39 105 18 ≤ ≤ NUM ejde-39 106 1 ‖u‖2w ‖u‖2w PROPN ejde-39 106 2 ≤ ≤ ADJ ejde-39 106 3 c1‖u‖21 c1‖u‖21 NOUN ejde-39 106 4 , , PUNCT ejde-39 106 5 ∀u ∀u NOUN ejde-39 106 6 ∈ ∈ PROPN ejde-39 106 7 w w NOUN ejde-39 106 8 , , PUNCT ejde-39 106 9 where where SCONJ ejde-39 106 10 0 0 NUM ejde-39 106 11 < < X ejde-39 106 12 c0 c0 X ejde-39 106 13 , , PUNCT ejde-39 106 14 c1 c1 PROPN ejde-39 106 15 < < X ejde-39 107 1 + + PROPN ejde-39 107 2 ∞ ∞ PROPN ejde-39 107 3 do do AUX ejde-39 107 4 not not PART ejde-39 107 5 depend depend VERB ejde-39 107 6 on on ADP ejde-39 107 7 u u PROPN ejde-39 107 8 ∈ ∈ PROPN ejde-39 107 9 w w NOUN ejde-39 107 10 . . PUNCT ejde-39 108 1 in in ADP ejde-39 108 2 addition addition NOUN ejde-39 108 3 , , PUNCT ejde-39 108 4 hλ(u hλ(u PROPN ejde-39 108 5 ) ) PUNCT ejde-39 108 6 : : PUNCT ejde-39 109 1 = = PUNCT ejde-39 109 2 ‖u‖2w ‖u‖2w NOUN ejde-39 109 3 − − NOUN ejde-39 109 4 λ|u|2l2 λ|u|2l2 NOUN ejde-39 109 5 = = SYM ejde-39 109 6 hλ(u+ hλ(u+ PROPN ejde-39 109 7 ) ) PUNCT ejde-39 110 1 + + NOUN ejde-39 110 2 hλ(u− hλ(u− X ejde-39 110 3 ) ) PUNCT ejde-39 110 4 = = SYM ejde-39 110 5 ‖u+‖21 ‖u+‖21 NOUN ejde-39 110 6 − − ADP ejde-39 110 7 ‖u−‖21 ‖u−‖21 ADJ ejde-39 110 8 , , PUNCT ejde-39 110 9 u u PRON ejde-39 110 10 ∈w ∈w NOUN ejde-39 110 11 . . PUNCT ejde-39 110 12 notice notice VERB ejde-39 110 13 that that SCONJ ejde-39 110 14 hλ(u hλ(u NOUN ejde-39 110 15 ) ) PUNCT ejde-39 111 1 = = PRON ejde-39 111 2 −‖u‖21 −‖u‖21 NOUN ejde-39 111 3 < < X ejde-39 111 4 0 0 PUNCT ejde-39 112 1 if if SCONJ ejde-39 112 2 u u PROPN ejde-39 112 3 ∈ ∈ PROPN ejde-39 112 4 w− w− NOUN ejde-39 112 5 \ \ X ejde-39 112 6 { { PUNCT ejde-39 112 7 0 0 NUM ejde-39 112 8 } } PUNCT ejde-39 112 9 , , PUNCT ejde-39 112 10 and and CCONJ ejde-39 112 11 hλ(u hλ(u NUM ejde-39 112 12 ) ) PUNCT ejde-39 113 1 = = NOUN ejde-39 113 2 ‖u‖21 ‖u‖21 NOUN ejde-39 113 3 > > X ejde-39 113 4 0 0 PUNCT ejde-39 114 1 if if SCONJ ejde-39 114 2 u u PROPN ejde-39 114 3 ∈w+ ∈w+ VERB ejde-39 114 4 \ \ PROPN ejde-39 114 5 { { PUNCT ejde-39 114 6 0 0 NUM ejde-39 114 7 } } PUNCT ejde-39 114 8 , , PUNCT ejde-39 114 9 for for ADP ejde-39 114 10 λ λ PROPN ejde-39 114 11 ∈ ∈ PROPN ejde-39 114 12 ( ( PUNCT ejde-39 114 13 λk λk ADP ejde-39 114 14 , , PUNCT ejde-39 114 15 , , PUNCT ejde-39 114 16 λk+1 λk+1 X ejde-39 114 17 ) ) PUNCT ejde-39 114 18 . . PUNCT ejde-39 115 1 with with ADP ejde-39 115 2 this this DET ejde-39 115 3 notation notation NOUN ejde-39 115 4 , , PUNCT ejde-39 115 5 we we PRON ejde-39 115 6 have have VERB ejde-39 115 7 eλ,µ(u eλ,µ(u NOUN ejde-39 115 8 ) ) PUNCT ejde-39 116 1 = = SYM ejde-39 116 2 1 1 NUM ejde-39 116 3 2 2 NUM ejde-39 116 4 hλ(u)− hλ(u)− ADP ejde-39 116 5 µ µ X ejde-39 116 6 q q X ejde-39 116 7 |u|qlq |u|qlq NUM ejde-39 116 8 − − PROPN ejde-39 117 1 ∫ ∫ PROPN ejde-39 118 1 g(x g(x NOUN ejde-39 118 2 , , PUNCT ejde-39 118 3 u)dx u)dx PROPN ejde-39 118 4 , , PUNCT ejde-39 118 5 reλ reλ NOUN ejde-39 118 6 ( ( PUNCT ejde-39 118 7 u u NOUN ejde-39 118 8 ) ) PUNCT ejde-39 118 9 = = SYM ejde-39 118 10 1 1 NUM ejde-39 118 11 2hλ(u)− 2hλ(u)− NUM ejde-39 118 12 ∫ ∫ NOUN ejde-39 118 13 g(x g(x NOUN ejde-39 118 14 , , PUNCT ejde-39 119 1 u)dx− u)dx− ADJ ejde-39 119 2 e e ADP ejde-39 119 3 1 1 NUM ejde-39 119 4 q q NOUN ejde-39 119 5 |u| |u| PROPN ejde-39 119 6 q q X ejde-39 119 7 lq lq INTJ ejde-39 119 8 , , PUNCT ejde-39 119 9 u u PRON ejde-39 119 10 ∈w ∈w VERB ejde-39 119 11 \ \ X ejde-39 119 12 { { PUNCT ejde-39 119 13 0 0 NUM ejde-39 119 14 } } PUNCT ejde-39 119 15 . . PUNCT ejde-39 120 1 obviously obviously ADV ejde-39 120 2 , , PUNCT ejde-39 120 3 re re VERB ejde-39 120 4 ∈ ∈ NOUN ejde-39 120 5 c1(w c1(w INTJ ejde-39 120 6 \ \ X ejde-39 120 7 { { PUNCT ejde-39 120 8 0},r 0},r NOUN ejde-39 120 9 ) ) PUNCT ejde-39 120 10 and and CCONJ ejde-39 120 11 deλ,µ(u deλ,µ(u PROPN ejde-39 120 12 ) ) PUNCT ejde-39 121 1 = = SYM ejde-39 121 2 0 0 NUM ejde-39 121 3 , , PUNCT ejde-39 121 4 eλ,µ(u eλ,µ(u NOUN ejde-39 121 5 ) ) PUNCT ejde-39 121 6 = = SYM ejde-39 121 7 e e X ejde-39 121 8 ⇔ ⇔ X ejde-39 121 9 dreλ dreλ NOUN ejde-39 121 10 ( ( PUNCT ejde-39 121 11 u u NOUN ejde-39 121 12 ) ) PUNCT ejde-39 121 13 = = SYM ejde-39 121 14 0 0 NUM ejde-39 121 15 , , PUNCT ejde-39 121 16 µ µ NOUN ejde-39 121 17 = = SYM ejde-39 121 18 reλ reλ NOUN ejde-39 121 19 ( ( PUNCT ejde-39 121 20 u u NOUN ejde-39 121 21 ) ) PUNCT ejde-39 121 22 , , PUNCT ejde-39 121 23 u u PRON ejde-39 121 24 ∈w ∈w VERB ejde-39 121 25 \ \ PROPN ejde-39 121 26 0 0 NUM ejde-39 121 27 . . PUNCT ejde-39 122 1 ( ( PUNCT ejde-39 122 2 2.1 2.1 NUM ejde-39 122 3 ) ) PUNCT ejde-39 122 4 ejde-2023/23 ejde-2023/23 NOUN ejde-39 122 5 prescribed prescribe VERB ejde-39 122 6 energy energy NOUN ejde-39 122 7 saddle saddle NOUN ejde-39 122 8 - - PUNCT ejde-39 122 9 point point NOUN ejde-39 122 10 solutions solution NOUN ejde-39 122 11 5 5 NUM ejde-39 122 12 to to PART ejde-39 122 13 avoid avoid VERB ejde-39 122 14 the the DET ejde-39 122 15 singularity singularity NOUN ejde-39 122 16 at at ADP ejde-39 122 17 origin origin NOUN ejde-39 122 18 of of ADP ejde-39 122 19 re re PROPN ejde-39 122 20 , , PUNCT ejde-39 122 21 we we PRON ejde-39 122 22 define define VERB ejde-39 122 23 φρ φρ NUM ejde-39 122 24 ∈ ∈ PROPN ejde-39 122 25 c∞(r c∞(r NOUN ejde-39 122 26 ) ) PUNCT ejde-39 122 27 , , PUNCT ejde-39 122 28 for for ADP ejde-39 122 29 ρ ρ PROPN ejde-39 122 30 > > X ejde-39 122 31 0 0 NUM ejde-39 122 32 such such ADJ ejde-39 122 33 that that PRON ejde-39 122 34 φρ(s φρ(s ADP ejde-39 122 35 ) ) PUNCT ejde-39 122 36 = = PRON ejde-39 122 37 { { PUNCT ejde-39 122 38 0 0 NUM ejde-39 122 39 if if SCONJ ejde-39 122 40 |s| |s| NOUN ejde-39 122 41 < < X ejde-39 122 42 ρ/2 ρ/2 PROPN ejde-39 122 43 , , PUNCT ejde-39 122 44 = = SYM ejde-39 122 45 1 1 NUM ejde-39 122 46 if if SCONJ ejde-39 122 47 |s| |s| NOUN ejde-39 122 48 > > X ejde-39 122 49 ρ ρ PROPN ejde-39 122 50 , , PUNCT ejde-39 122 51 and and CCONJ ejde-39 122 52 introduce introduce VERB ejde-39 122 53 reρ reρ NOUN ejde-39 122 54 ( ( PUNCT ejde-39 122 55 u u NOUN ejde-39 122 56 ) ) PUNCT ejde-39 122 57 = = SYM ejde-39 122 58 { { PUNCT ejde-39 122 59 φρ(‖u‖1)re(u φρ(‖u‖1)re(u NOUN ejde-39 122 60 ) ) PUNCT ejde-39 122 61 , , PUNCT ejde-39 122 62 u u PRON ejde-39 122 63 ∈w ∈w VERB ejde-39 122 64 \ \ PROPN ejde-39 122 65 0 0 NUM ejde-39 122 66 , , PUNCT ejde-39 122 67 0 0 NUM ejde-39 122 68 , , PUNCT ejde-39 122 69 u u NOUN ejde-39 122 70 = = NOUN ejde-39 122 71 0 0 NUM ejde-39 122 72 . . PUNCT ejde-39 123 1 thus thus ADV ejde-39 123 2 , , PUNCT ejde-39 123 3 reρ reρ X ejde-39 123 4 ( ( PUNCT ejde-39 123 5 u u NOUN ejde-39 123 6 ) ) PUNCT ejde-39 123 7 ∈ ∈ PROPN ejde-39 123 8 c1(w c1(w NOUN ejde-39 123 9 ) ) PUNCT ejde-39 123 10 for for ADP ejde-39 123 11 any any DET ejde-39 123 12 ρ ρ NOUN ejde-39 123 13 > > X ejde-39 123 14 0 0 NUM ejde-39 123 15 . . PUNCT ejde-39 124 1 we we PRON ejde-39 124 2 define define VERB ejde-39 124 3 br br NOUN ejde-39 124 4 : : PUNCT ejde-39 124 5 = = SYM ejde-39 124 6 { { PUNCT ejde-39 124 7 u u NOUN ejde-39 124 8 ∈w ∈w VERB ejde-39 124 9 : : PUNCT ejde-39 124 10 ‖u‖1 ‖u‖1 VERB ejde-39 124 11 ≤ ≤ NOUN ejde-39 124 12 r r NOUN ejde-39 124 13 } } PUNCT ejde-39 124 14 , , PUNCT ejde-39 124 15 r r NOUN ejde-39 124 16 > > X ejde-39 124 17 0 0 NUM ejde-39 124 18 . . PUNCT ejde-39 125 1 we we PRON ejde-39 125 2 need need VERB ejde-39 125 3 the the DET ejde-39 125 4 following follow VERB ejde-39 125 5 result result NOUN ejde-39 125 6 . . PUNCT ejde-39 126 1 lemma lemma PROPN ejde-39 126 2 2.1 2.1 NUM ejde-39 126 3 . . PUNCT ejde-39 126 4 assume assume VERB ejde-39 126 5 that that SCONJ ejde-39 126 6 e e NOUN ejde-39 126 7 > > X ejde-39 126 8 0 0 PUNCT ejde-39 127 1 and and CCONJ ejde-39 127 2 λ λ X ejde-39 127 3 ∈ ∈ PROPN ejde-39 127 4 ( ( PUNCT ejde-39 127 5 λk λk ADP ejde-39 127 6 , , PUNCT ejde-39 127 7 , , PUNCT ejde-39 127 8 λk+1 λk+1 X ejde-39 127 9 ) ) PUNCT ejde-39 127 10 . . PUNCT ejde-39 128 1 then then ADV ejde-39 128 2 there there PRON ejde-39 128 3 exists exist VERB ejde-39 128 4 ρ(e ρ(e PROPN ejde-39 128 5 ) ) PUNCT ejde-39 128 6 > > X ejde-39 128 7 0 0 PUNCT ejde-39 128 8 such such ADJ ejde-39 128 9 that that DET ejde-39 128 10 re(u re(u NOUN ejde-39 128 11 ) ) PUNCT ejde-39 128 12 < < X ejde-39 128 13 0 0 NUM ejde-39 128 14 for for SCONJ ejde-39 128 15 any any DET ejde-39 128 16 u u NOUN ejde-39 128 17 ∈ ∈ PROPN ejde-39 128 18 bρ bρ VERB ejde-39 128 19 with with ADP ejde-39 128 20 0 0 NUM ejde-39 128 21 < < X ejde-39 128 22 ρ ρ X ejde-39 128 23 < < X ejde-39 128 24 ρ(e ρ(e PROPN ejde-39 128 25 ) ) PUNCT ejde-39 128 26 . . PUNCT ejde-39 129 1 proof proof NOUN ejde-39 129 2 . . PUNCT ejde-39 130 1 since since SCONJ ejde-39 130 2 g(x g(x NOUN ejde-39 130 3 , , PUNCT ejde-39 130 4 u u NOUN ejde-39 130 5 ) ) PUNCT ejde-39 130 6 ≥ ≥ NOUN ejde-39 130 7 0 0 NUM ejde-39 130 8 a.e a.e PROPN ejde-39 130 9 . . PROPN ejde-39 130 10 ω ω PROPN ejde-39 130 11 , , PUNCT ejde-39 130 12 u u PROPN ejde-39 130 13 ∈ ∈ PROPN ejde-39 130 14 r r NOUN ejde-39 130 15 , , PUNCT ejde-39 130 16 reλ reλ NOUN ejde-39 130 17 ( ( PUNCT ejde-39 130 18 u u NOUN ejde-39 130 19 ) ) PUNCT ejde-39 130 20 < < X ejde-39 130 21 q q PROPN ejde-39 130 22 1 1 NUM ejde-39 130 23 |u|qlq |u|qlq NUM ejde-39 130 24 ( ( PUNCT ejde-39 130 25 1 1 NUM ejde-39 130 26 2 2 NUM ejde-39 130 27 hλ(u)− hλ(u)− NOUN ejde-39 130 28 e e NOUN ejde-39 130 29 ) ) PUNCT ejde-39 130 30 < < X ejde-39 130 31 q q X ejde-39 130 32 |u|qlq |u|qlq X ejde-39 130 33 ( ( PUNCT ejde-39 130 34 1 1 NUM ejde-39 130 35 2 2 NUM ejde-39 130 36 ‖u‖21 ‖u‖21 NOUN ejde-39 130 37 − − PROPN ejde-39 130 38 e e NOUN ejde-39 130 39 ) ) PUNCT ejde-39 130 40 , , PUNCT ejde-39 130 41 u u PRON ejde-39 130 42 ∈w ∈w VERB ejde-39 130 43 \ \ X ejde-39 130 44 { { PUNCT ejde-39 130 45 0 0 NUM ejde-39 130 46 } } PUNCT ejde-39 130 47 . . PUNCT ejde-39 131 1 hence hence ADV ejde-39 131 2 , , PUNCT ejde-39 131 3 setting set VERB ejde-39 131 4 ρ(e ρ(e PROPN ejde-39 131 5 ) ) PUNCT ejde-39 131 6 : : PUNCT ejde-39 132 1 = = PUNCT ejde-39 132 2 √ √ NUM ejde-39 132 3 2e 2e NOUN ejde-39 132 4 we we PRON ejde-39 132 5 obtain obtain VERB ejde-39 132 6 the the DET ejde-39 132 7 proof proof NOUN ejde-39 132 8 . . PUNCT ejde-39 133 1 � � PROPN ejde-39 133 2 corollary corollary PROPN ejde-39 133 3 2.2 2.2 NUM ejde-39 133 4 . . PUNCT ejde-39 134 1 assume assume VERB ejde-39 134 2 that that SCONJ ejde-39 134 3 ρ ρ PROPN ejde-39 134 4 < < X ejde-39 134 5 ρ(e ρ(e PROPN ejde-39 134 6 ) ) PUNCT ejde-39 134 7 . . PUNCT ejde-39 135 1 if if SCONJ ejde-39 135 2 û û NUM ejde-39 135 3 is be AUX ejde-39 135 4 a a DET ejde-39 135 5 critical critical ADJ ejde-39 135 6 point point NOUN ejde-39 135 7 of of ADP ejde-39 135 8 reρ reρ NOUN ejde-39 135 9 ( ( PUNCT ejde-39 135 10 u u NOUN ejde-39 135 11 ) ) PUNCT ejde-39 135 12 such such ADJ ejde-39 135 13 that that DET ejde-39 135 14 reρ reρ NOUN ejde-39 135 15 ( ( PUNCT ejde-39 135 16 û û X ejde-39 135 17 ) ) PUNCT ejde-39 135 18 > > X ejde-39 135 19 0 0 NUM ejde-39 135 20 , , PUNCT ejde-39 135 21 then then ADV ejde-39 135 22 u u NOUN ejde-39 135 23 is be AUX ejde-39 135 24 a a DET ejde-39 135 25 critical critical ADJ ejde-39 135 26 point point NOUN ejde-39 135 27 of of ADP ejde-39 135 28 re(u re(u NOUN ejde-39 135 29 ) ) PUNCT ejde-39 135 30 as as ADV ejde-39 135 31 well well ADV ejde-39 135 32 . . PUNCT ejde-39 136 1 proof proof NOUN ejde-39 136 2 . . PUNCT ejde-39 137 1 by by ADP ejde-39 137 2 lemma lemma PROPN ejde-39 137 3 2.1 2.1 NUM ejde-39 137 4 , , PUNCT ejde-39 137 5 reρ reρ X ejde-39 137 6 ( ( PUNCT ejde-39 137 7 û û X ejde-39 137 8 ) ) PUNCT ejde-39 137 9 > > SYM ejde-39 137 10 0 0 NUM ejde-39 137 11 implies imply VERB ejde-39 137 12 that that SCONJ ejde-39 137 13 ‖û‖w ‖û‖w PROPN ejde-39 137 14 ≥ ≥ PROPN ejde-39 137 15 ρ ρ NOUN ejde-39 137 16 . . PUNCT ejde-39 138 1 therefore therefore ADV ejde-39 138 2 reρ reρ X ejde-39 138 3 ( ( PUNCT ejde-39 138 4 û û X ejde-39 138 5 ) ) PUNCT ejde-39 138 6 = = SYM ejde-39 138 7 re(û re(û NOUN ejde-39 138 8 ) ) PUNCT ejde-39 138 9 = = SYM ejde-39 138 10 µ µ X ejde-39 138 11 and and CCONJ ejde-39 138 12 dre(û dre(û NOUN ejde-39 138 13 ) ) PUNCT ejde-39 138 14 = = SYM ejde-39 139 1 0 0 X ejde-39 139 2 . . X ejde-39 139 3 � � PROPN ejde-39 139 4 we we PRON ejde-39 139 5 say say VERB ejde-39 139 6 that that SCONJ ejde-39 139 7 ( ( PUNCT ejde-39 139 8 un un PROPN ejde-39 139 9 ) ) PUNCT ejde-39 139 10 ⊂ ⊂ PROPN ejde-39 139 11 w w PROPN ejde-39 139 12 is be AUX ejde-39 139 13 a a DET ejde-39 139 14 cerami cerami NOUN ejde-39 139 15 sequence sequence NOUN ejde-39 139 16 at at ADP ejde-39 139 17 the the DET ejde-39 139 18 level level NOUN ejde-39 139 19 c c NOUN ejde-39 139 20 ∈ ∈ NOUN ejde-39 139 21 r r NOUN ejde-39 139 22 of of ADP ejde-39 139 23 re re NOUN ejde-39 139 24 , , PUNCT ejde-39 139 25 in in ADP ejde-39 139 26 short short ADJ ejde-39 139 27 ( ( PUNCT ejde-39 139 28 ce ce PROPN ejde-39 139 29 ) ) PUNCT ejde-39 139 30 sequence sequence NOUN ejde-39 139 31 , , PUNCT ejde-39 140 1 whenever whenever SCONJ ejde-39 140 2 re(un)→ re(un)→ PROPN ejde-39 140 3 c c PROPN ejde-39 140 4 and and CCONJ ejde-39 140 5 ( ( PUNCT ejde-39 140 6 1+‖un‖w 1+‖un‖w NUM ejde-39 140 7 ) ) PUNCT ejde-39 140 8 ‖dre(un)‖ ‖dre(un)‖ NOUN ejde-39 140 9 → → SYM ejde-39 140 10 0 0 NUM ejde-39 140 11 as as ADP ejde-39 140 12 n→∞. n→∞. PROPN ejde-39 140 13 the the DET ejde-39 140 14 functional functional ADJ ejde-39 140 15 re re NOUN ejde-39 140 16 satisfies satisfy VERB ejde-39 140 17 the the DET ejde-39 140 18 cerami cerami PROPN ejde-39 140 19 condition condition NOUN ejde-39 140 20 at at ADP ejde-39 140 21 the the DET ejde-39 140 22 level level NOUN ejde-39 140 23 c c NOUN ejde-39 140 24 ∈ ∈ NOUN ejde-39 140 25 r r NOUN ejde-39 140 26 , , PUNCT ejde-39 140 27 in in ADP ejde-39 140 28 short short ADJ ejde-39 140 29 ( ( PUNCT ejde-39 140 30 ce ce NOUN ejde-39 140 31 ) ) PUNCT ejde-39 140 32 condition condition NOUN ejde-39 140 33 , , PUNCT ejde-39 140 34 whenever whenever SCONJ ejde-39 140 35 any any DET ejde-39 140 36 ( ( PUNCT ejde-39 140 37 ce ce NOUN ejde-39 140 38 ) ) PUNCT ejde-39 140 39 sequence sequence NOUN ejde-39 140 40 possesses possess VERB ejde-39 140 41 a a DET ejde-39 140 42 convergent convergent NOUN ejde-39 140 43 subsequence subsequence NOUN ejde-39 140 44 . . PUNCT ejde-39 141 1 the the DET ejde-39 141 2 definitions definition NOUN ejde-39 141 3 of of ADP ejde-39 141 4 the the DET ejde-39 141 5 ( ( PUNCT ejde-39 141 6 ce ce PROPN ejde-39 141 7 ) ) PUNCT ejde-39 141 8 sequence sequence NOUN ejde-39 141 9 and and CCONJ ejde-39 141 10 the the DET ejde-39 141 11 ( ( PUNCT ejde-39 141 12 ce ce PROPN ejde-39 141 13 ) ) PUNCT ejde-39 141 14 condition condition NOUN ejde-39 141 15 for for ADP ejde-39 141 16 reρ reρ NOUN ejde-39 141 17 are be AUX ejde-39 141 18 similar similar ADJ ejde-39 141 19 . . PUNCT ejde-39 142 1 corollary corollary ADJ ejde-39 142 2 2.3 2.3 NUM ejde-39 142 3 . . PUNCT ejde-39 143 1 if if SCONJ ejde-39 143 2 ρ ρ PROPN ejde-39 143 3 < < X ejde-39 143 4 ρ(e ρ(e PROPN ejde-39 143 5 ) ) PUNCT ejde-39 143 6 , , PUNCT ejde-39 143 7 then then ADV ejde-39 143 8 reρ reρ NOUN ejde-39 143 9 ( ( PUNCT ejde-39 143 10 u u NOUN ejde-39 143 11 ) ) PUNCT ejde-39 143 12 satisfies satisfy VERB ejde-39 143 13 the the DET ejde-39 143 14 ( ( PUNCT ejde-39 143 15 ce ce PROPN ejde-39 143 16 ) ) PUNCT ejde-39 143 17 condition condition NOUN ejde-39 143 18 at at ADP ejde-39 143 19 level level NOUN ejde-39 143 20 c c NOUN ejde-39 143 21 > > X ejde-39 143 22 0 0 PUNCT ejde-39 144 1 if if SCONJ ejde-39 144 2 and and CCONJ ejde-39 144 3 only only ADV ejde-39 144 4 if if SCONJ ejde-39 144 5 re(u re(u NOUN ejde-39 144 6 ) ) PUNCT ejde-39 144 7 does do VERB ejde-39 144 8 . . PUNCT ejde-39 145 1 proof proof NOUN ejde-39 145 2 . . PUNCT ejde-39 146 1 assume assume VERB ejde-39 146 2 re(u re(u X ejde-39 146 3 ) ) PUNCT ejde-39 146 4 satisfies satisfy VERB ejde-39 146 5 the the DET ejde-39 146 6 ( ( PUNCT ejde-39 146 7 ce ce PROPN ejde-39 146 8 ) ) PUNCT ejde-39 146 9 condition condition NOUN ejde-39 146 10 at at ADP ejde-39 146 11 c c PROPN ejde-39 146 12 > > X ejde-39 146 13 0 0 X ejde-39 146 14 . . PUNCT ejde-39 147 1 let let AUX ejde-39 147 2 ( ( PUNCT ejde-39 147 3 un un VERB ejde-39 147 4 ) ) PUNCT ejde-39 147 5 be be VERB ejde-39 147 6 a a DET ejde-39 147 7 ( ( PUNCT ejde-39 147 8 ce ce NOUN ejde-39 147 9 ) ) PUNCT ejde-39 147 10 sequence sequence NOUN ejde-39 147 11 forreρ forreρ NOUN ejde-39 147 12 at at ADP ejde-39 147 13 c c PROPN ejde-39 147 14 , , PUNCT ejde-39 147 15 i.e. i.e. X ejde-39 147 16 , , PUNCT ejde-39 147 17 reρ reρ X ejde-39 147 18 ( ( PUNCT ejde-39 147 19 un)→ un)→ ADP ejde-39 147 20 c c PROPN ejde-39 147 21 and and CCONJ ejde-39 147 22 ( ( PUNCT ejde-39 147 23 1+‖un‖1)‖dreρ 1+‖un‖1)‖dreρ NUM ejde-39 147 24 ( ( PUNCT ejde-39 147 25 un)‖∗ un)‖∗ NOUN ejde-39 147 26 → → SYM ejde-39 147 27 0 0 NUM ejde-39 147 28 as as ADP ejde-39 147 29 n→∞. n→∞. VERB ejde-39 147 30 by by ADP ejde-39 147 31 lemma lemma PROPN ejde-39 147 32 2.1 2.1 NUM ejde-39 147 33 , , PUNCT ejde-39 147 34 reρ reρ X ejde-39 147 35 ( ( PUNCT ejde-39 147 36 u u NOUN ejde-39 147 37 ) ) PUNCT ejde-39 147 38 ≤ ≤ NOUN ejde-39 147 39 0 0 PUNCT ejde-39 148 1 as as SCONJ ejde-39 148 2 u u PROPN ejde-39 148 3 ∈ ∈ PROPN ejde-39 148 4 bρ bρ VERB ejde-39 148 5 , , PUNCT ejde-39 148 6 whereas whereas SCONJ ejde-39 148 7 reρ reρ X ejde-39 148 8 ( ( PUNCT ejde-39 148 9 u u NOUN ejde-39 148 10 ) ) PUNCT ejde-39 148 11 = = SYM ejde-39 148 12 re(u re(u NOUN ejde-39 148 13 ) ) PUNCT ejde-39 148 14 for for ADP ejde-39 148 15 u u PROPN ejde-39 148 16 ∈ ∈ PROPN ejde-39 148 17 w w PROPN ejde-39 148 18 \ \ PROPN ejde-39 148 19 bρ bρ PROPN ejde-39 148 20 . . PUNCT ejde-39 149 1 thus thus ADV ejde-39 149 2 , , PUNCT ejde-39 149 3 reρ reρ X ejde-39 149 4 ( ( PUNCT ejde-39 149 5 u u NOUN ejde-39 149 6 ) ) PUNCT ejde-39 149 7 > > SYM ejde-39 149 8 0 0 NUM ejde-39 149 9 implies imply VERB ejde-39 149 10 re(u re(u X ejde-39 149 11 ) ) PUNCT ejde-39 150 1 = = SYM ejde-39 150 2 reρ reρ X ejde-39 150 3 ( ( PUNCT ejde-39 150 4 u u NOUN ejde-39 150 5 ) ) PUNCT ejde-39 150 6 and and CCONJ ejde-39 150 7 dre(u dre(u PROPN ejde-39 150 8 ) ) PUNCT ejde-39 151 1 = = SYM ejde-39 151 2 dreρ dreρ X ejde-39 151 3 ( ( PUNCT ejde-39 151 4 u u NOUN ejde-39 151 5 ) ) PUNCT ejde-39 151 6 . . PUNCT ejde-39 152 1 therefore therefore ADV ejde-39 152 2 ( ( PUNCT ejde-39 152 3 un un PROPN ejde-39 152 4 ) ) PUNCT ejde-39 152 5 is be AUX ejde-39 152 6 also also ADV ejde-39 152 7 ( ( PUNCT ejde-39 152 8 ce ce PROPN ejde-39 152 9 ) ) PUNCT ejde-39 152 10 sequence sequence NOUN ejde-39 152 11 for for ADP ejde-39 152 12 re re NOUN ejde-39 152 13 and and CCONJ ejde-39 152 14 consequently consequently ADV ejde-39 152 15 , , PUNCT ejde-39 152 16 ( ( PUNCT ejde-39 152 17 un un PROPN ejde-39 152 18 ) ) PUNCT ejde-39 152 19 possesses possess VERB ejde-39 152 20 a a DET ejde-39 152 21 convergent convergent NOUN ejde-39 152 22 subsequence subsequence NOUN ejde-39 152 23 in in ADP ejde-39 152 24 w w PROPN ejde-39 152 25 . . PUNCT ejde-39 153 1 the the DET ejde-39 153 2 proof proof NOUN ejde-39 153 3 of of ADP ejde-39 153 4 opposite opposite ADJ ejde-39 153 5 statement statement NOUN ejde-39 153 6 is be AUX ejde-39 153 7 similar similar ADJ ejde-39 153 8 . . PUNCT ejde-39 154 1 � � PROPN ejde-39 154 2 3 3 NUM ejde-39 154 3 . . PUNCT ejde-39 155 1 properties property NOUN ejde-39 155 2 of of ADP ejde-39 155 3 reρ reρ NOUN ejde-39 155 4 consider consider VERB ejde-39 155 5 the the DET ejde-39 155 6 sphere sphere NOUN ejde-39 155 7 s±r s±r PROPN ejde-39 155 8 : : PUNCT ejde-39 155 9 = = SYM ejde-39 155 10 { { PUNCT ejde-39 155 11 u u NOUN ejde-39 155 12 ∈ ∈ PROPN ejde-39 155 13 w± w± NOUN ejde-39 155 14 : : PUNCT ejde-39 155 15 ‖u‖1 ‖u‖1 VERB ejde-39 155 16 = = SYM ejde-39 156 1 r r X ejde-39 156 2 } } PUNCT ejde-39 156 3 , , PUNCT ejde-39 156 4 r r NOUN ejde-39 156 5 > > X ejde-39 156 6 0 0 X ejde-39 156 7 . . PUNCT ejde-39 156 8 recall recall VERB ejde-39 156 9 that that SCONJ ejde-39 156 10 by by ADP ejde-39 156 11 the the DET ejde-39 156 12 assumption assumption NOUN ejde-39 156 13 g(x g(x PROPN ejde-39 156 14 , , PUNCT ejde-39 156 15 u u NOUN ejde-39 156 16 ) ) PUNCT ejde-39 156 17 ≤ ≤ NOUN ejde-39 156 18 c(|u|γ1−1 c(|u|γ1−1 VERB ejde-39 156 19 + + CCONJ ejde-39 156 20 |u|γ2−1 |u|γ2−1 PROPN ejde-39 156 21 ) ) PUNCT ejde-39 156 22 a.e a.e PROPN ejde-39 156 23 . . PROPN ejde-39 156 24 ω ω PROPN ejde-39 156 25 , , PUNCT ejde-39 156 26 u u PROPN ejde-39 156 27 ∈ ∈ PROPN ejde-39 156 28 r r NOUN ejde-39 156 29 , , PUNCT ejde-39 156 30 for for ADP ejde-39 156 31 some some DET ejde-39 156 32 γ1 γ1 NOUN ejde-39 156 33 , , PUNCT ejde-39 156 34 γ2 γ2 PROPN ejde-39 156 35 ∈ ∈ PROPN ejde-39 156 36 ( ( PUNCT ejde-39 156 37 2 2 NUM ejde-39 156 38 , , PUNCT ejde-39 156 39 2∗ 2∗ NUM ejde-39 156 40 ) ) PUNCT ejde-39 156 41 , , PUNCT ejde-39 156 42 c c X ejde-39 156 43 > > X ejde-39 156 44 0 0 X ejde-39 156 45 . . PUNCT ejde-39 157 1 hence hence ADV ejde-39 157 2 , , PUNCT ejde-39 157 3 for for ADP ejde-39 157 4 any any DET ejde-39 157 5 given give VERB ejde-39 157 6 ε ε PROPN ejde-39 157 7 > > X ejde-39 157 8 0 0 PROPN ejde-39 157 9 , , PUNCT ejde-39 157 10 there there PRON ejde-39 157 11 exist exist VERB ejde-39 157 12 c(ε c(ε PROPN ejde-39 157 13 ) ) PUNCT ejde-39 157 14 > > X ejde-39 157 15 0 0 PUNCT ejde-39 158 1 such such ADJ ejde-39 158 2 that that SCONJ ejde-39 158 3 g(x g(x NOUN ejde-39 158 4 , , PUNCT ejde-39 158 5 s s NOUN ejde-39 158 6 ) ) PUNCT ejde-39 158 7 ≤ ≤ NUM ejde-39 158 8 ε ε PROPN ejde-39 158 9 2 2 NUM ejde-39 158 10 |s|2 |s|2 NOUN ejde-39 158 11 + + CCONJ ejde-39 158 12 c(ε)|s|γ c(ε)|s|γ PROPN ejde-39 158 13 , , PUNCT ejde-39 158 14 ∀s ∀s PROPN ejde-39 158 15 ∈ ∈ PROPN ejde-39 158 16 r r PROPN ejde-39 158 17 , , PUNCT ejde-39 158 18 a.e a.e PROPN ejde-39 158 19 . . PROPN ejde-39 158 20 ω ω PROPN ejde-39 158 21 , , PUNCT ejde-39 158 22 where where SCONJ ejde-39 158 23 γ γ X ejde-39 158 24 : : PUNCT ejde-39 158 25 = = SYM ejde-39 158 26 max{γ1 max{γ1 NOUN ejde-39 158 27 , , PUNCT ejde-39 158 28 γ2 γ2 NOUN ejde-39 158 29 } } PUNCT ejde-39 158 30 . . PUNCT ejde-39 159 1 this this PRON ejde-39 159 2 by by ADP ejde-39 159 3 the the DET ejde-39 159 4 sobolev sobolev PROPN ejde-39 159 5 inequalities inequality NOUN ejde-39 159 6 implies∫ implies∫ VERB ejde-39 159 7 g(x g(x NOUN ejde-39 159 8 , , PUNCT ejde-39 159 9 u)dx u)dx PROPN ejde-39 159 10 ≤ ≤ NUM ejde-39 159 11 ε ε NOUN ejde-39 159 12 2 2 NUM ejde-39 159 13 c1‖u‖21 c1‖u‖21 NOUN ejde-39 159 14 + + CCONJ ejde-39 159 15 c2(ε)‖u‖γ1 c2(ε)‖u‖γ1 NOUN ejde-39 159 16 , , PUNCT ejde-39 159 17 u u PRON ejde-39 159 18 ∈w ∈w NOUN ejde-39 159 19 , , PUNCT ejde-39 159 20 ( ( PUNCT ejde-39 159 21 3.1 3.1 NUM ejde-39 159 22 ) ) PUNCT ejde-39 159 23 6 6 NUM ejde-39 159 24 y. y. NOUN ejde-39 159 25 il’yasov il’yasov PROPN ejde-39 159 26 , , PUNCT ejde-39 159 27 e. e. PROPN ejde-39 159 28 d. d. PROPN ejde-39 159 29 silva silva PROPN ejde-39 159 30 , , PUNCT ejde-39 159 31 m. m. NOUN ejde-39 159 32 l. l. PROPN ejde-39 159 33 silva silva PROPN ejde-39 159 34 ejde-2023/23 ejde-2023/23 PROPN ejde-39 159 35 where where SCONJ ejde-39 159 36 c1 c1 PROPN ejde-39 159 37 , , PUNCT ejde-39 159 38 c2(ε c2(ε PROPN ejde-39 159 39 ) ) PUNCT ejde-39 159 40 ∈ ∈ PROPN ejde-39 159 41 ( ( PUNCT ejde-39 159 42 0,+∞ 0,+∞ NUM ejde-39 159 43 ) ) PUNCT ejde-39 159 44 do do AUX ejde-39 159 45 not not PART ejde-39 159 46 depend depend VERB ejde-39 159 47 on on ADP ejde-39 159 48 u u PROPN ejde-39 159 49 ∈ ∈ PROPN ejde-39 159 50 w w NOUN ejde-39 159 51 and and CCONJ ejde-39 159 52 c1 c1 PROPN ejde-39 159 53 does do AUX ejde-39 159 54 not not PART ejde-39 159 55 depend depend VERB ejde-39 159 56 on on ADP ejde-39 159 57 ε ε PROPN ejde-39 159 58 > > X ejde-39 159 59 0 0 PROPN ejde-39 159 60 . . PUNCT ejde-39 160 1 proposition proposition NOUN ejde-39 160 2 3.1 3.1 NUM ejde-39 160 3 . . PUNCT ejde-39 161 1 for for ADP ejde-39 161 2 any any DET ejde-39 161 3 λ λ PROPN ejde-39 161 4 ∈ ∈ PROPN ejde-39 161 5 ( ( PUNCT ejde-39 161 6 λk λk X ejde-39 161 7 , , PUNCT ejde-39 161 8 λk+1 λk+1 NUM ejde-39 161 9 ) ) PUNCT ejde-39 161 10 , , PUNCT ejde-39 161 11 there there PRON ejde-39 161 12 exist exist VERB ejde-39 161 13 ekλ ekλ PROPN ejde-39 161 14 > > X ejde-39 161 15 0 0 PUNCT ejde-39 162 1 and and CCONJ ejde-39 162 2 rkλ rkλ NOUN ejde-39 162 3 > > X ejde-39 162 4 0 0 NUM ejde-39 163 1 such such ADJ ejde-39 163 2 that that SCONJ ejde-39 163 3 infw∈s+ infw∈s+ NOUN ejde-39 163 4 rk rk NOUN ejde-39 163 5 λ λ PROPN ejde-39 163 6 reρ reρ X ejde-39 163 7 ( ( PUNCT ejde-39 163 8 w w NOUN ejde-39 163 9 ) ) PUNCT ejde-39 163 10 > > X ejde-39 163 11 0 0 NUM ejde-39 163 12 , , PUNCT ejde-39 163 13 for for ADP ejde-39 163 14 any any DET ejde-39 163 15 e e NOUN ejde-39 163 16 ∈ ∈ PROPN ejde-39 163 17 [ [ X ejde-39 163 18 0 0 NUM ejde-39 163 19 , , PUNCT ejde-39 163 20 ekλ ekλ ADJ ejde-39 163 21 ) ) PUNCT ejde-39 163 22 , , PUNCT ejde-39 163 23 for for ADP ejde-39 163 24 all all DET ejde-39 163 25 ρ ρ NOUN ejde-39 163 26 ∈ ∈ PROPN ejde-39 163 27 ( ( PUNCT ejde-39 163 28 0 0 NUM ejde-39 163 29 , , PUNCT ejde-39 163 30 rkλ rkλ NOUN ejde-39 163 31 ) ) PUNCT ejde-39 163 32 . . PUNCT ejde-39 164 1 proof proof NOUN ejde-39 164 2 . . PUNCT ejde-39 165 1 note note VERB ejde-39 165 2 that that SCONJ ejde-39 165 3 hλ(w hλ(w X ejde-39 165 4 ) ) PUNCT ejde-39 165 5 = = SYM ejde-39 165 6 ‖w‖21 ‖w‖21 NOUN ejde-39 165 7 , , PUNCT ejde-39 165 8 ∀w ∀w PROPN ejde-39 165 9 ∈ ∈ PROPN ejde-39 165 10 w+ w+ NOUN ejde-39 165 11 . . PUNCT ejde-39 166 1 take take VERB ejde-39 166 2 ε ε PROPN ejde-39 166 3 ∈ ∈ PROPN ejde-39 166 4 ( ( PUNCT ejde-39 166 5 0 0 NUM ejde-39 166 6 , , PUNCT ejde-39 166 7 1 1 NUM ejde-39 166 8 / / SYM ejde-39 166 9 c1 c1 NOUN ejde-39 166 10 ) ) PUNCT ejde-39 166 11 . . PUNCT ejde-39 167 1 then then ADV ejde-39 167 2 by by ADP ejde-39 167 3 ( ( PUNCT ejde-39 167 4 3.1 3.1 NUM ejde-39 167 5 ) ) PUNCT ejde-39 167 6 we we PRON ejde-39 167 7 have have VERB ejde-39 167 8 re(w re(w ADV ejde-39 167 9 ) ) PUNCT ejde-39 167 10 ≥ ≥ NOUN ejde-39 168 1 q q NOUN ejde-39 168 2 1 1 NUM ejde-39 168 3 2 2 NUM ejde-39 168 4 ( ( PUNCT ejde-39 168 5 1− 1− NUM ejde-39 168 6 c1ε c1ε NOUN ejde-39 168 7 ) ) PUNCT ejde-39 168 8 ‖w‖21 ‖w‖21 NOUN ejde-39 168 9 − − PROPN ejde-39 168 10 c2(ε)‖w‖γ1 c2(ε)‖w‖γ1 PROPN ejde-39 169 1 − − PROPN ejde-39 169 2 e e NOUN ejde-39 169 3 |w|qlq |w|qlq ADP ejde-39 169 4 = = SYM ejde-39 169 5 q q NOUN ejde-39 169 6 f f X ejde-39 169 7 ( ( PUNCT ejde-39 169 8 ‖w‖1)− ‖w‖1)− NOUN ejde-39 169 9 e e NOUN ejde-39 169 10 |w|qlq |w|qlq X ejde-39 169 11 , , PUNCT ejde-39 169 12 ∀w ∀w PROPN ejde-39 169 13 ∈w+ ∈w+ PROPN ejde-39 169 14 , , PUNCT ejde-39 169 15 where where SCONJ ejde-39 169 16 f(r f(r NOUN ejde-39 169 17 ) ) PUNCT ejde-39 169 18 : : PUNCT ejde-39 170 1 = = SYM ejde-39 170 2 1 1 NUM ejde-39 170 3 2 2 NUM ejde-39 170 4 ( ( PUNCT ejde-39 170 5 1−c1ε)r 1−c1ε)r NUM ejde-39 170 6 2−c2(ε)rγ 2−c2(ε)rγ NUM ejde-39 170 7 . . PUNCT ejde-39 171 1 observe observe VERB ejde-39 171 2 that that SCONJ ejde-39 171 3 f(r f(r NOUN ejde-39 171 4 ) ) PUNCT ejde-39 171 5 attains attain VERB ejde-39 171 6 its its PRON ejde-39 171 7 global global ADJ ejde-39 171 8 maximum maximum PROPN ejde-39 171 9 ekλ ekλ PROPN ejde-39 171 10 : : PUNCT ejde-39 171 11 = = PUNCT ejde-39 171 12 f(rkλ f(rkλ NOUN ejde-39 171 13 ) ) PUNCT ejde-39 171 14 at at ADP ejde-39 171 15 rkλ rkλ NOUN ejde-39 171 16 : : PUNCT ejde-39 171 17 = = SYM ejde-39 172 1 [ [ X ejde-39 172 2 ( ( PUNCT ejde-39 172 3 1− 1− NUM ejde-39 172 4 c1ε)/(γc2(ε c1ε)/(γc2(ε PROPN ejde-39 172 5 ) ) PUNCT ejde-39 172 6 ) ) PUNCT ejde-39 172 7 ] ] PUNCT ejde-39 173 1 1/(γ−2 1/(γ−2 NUM ejde-39 173 2 ) ) PUNCT ejde-39 173 3 . . PUNCT ejde-39 174 1 thus thus ADV ejde-39 174 2 , , PUNCT ejde-39 174 3 for for ADP ejde-39 174 4 any any DET ejde-39 174 5 e e NOUN ejde-39 174 6 ∈ ∈ PROPN ejde-39 175 1 [ [ X ejde-39 175 2 0 0 NUM ejde-39 175 3 , , PUNCT ejde-39 175 4 ekλ ekλ ADJ ejde-39 175 5 ) ) PUNCT ejde-39 175 6 , , PUNCT ejde-39 175 7 inf inf PROPN ejde-39 175 8 w∈s+ w∈s+ PROPN ejde-39 175 9 rk rk NOUN ejde-39 175 10 λ λ PROPN ejde-39 175 11 re(w re(w PRON ejde-39 175 12 ) ) PUNCT ejde-39 175 13 ≥ ≥ PROPN ejde-39 175 14 inf inf PROPN ejde-39 175 15 w∈s+ w∈s+ PROPN ejde-39 175 16 rk rk NOUN ejde-39 175 17 λ λ X ejde-39 175 18 q q NOUN ejde-39 175 19 f(rkλ)− f(rkλ)− NUM ejde-39 175 20 e e X ejde-39 175 21 |w|qlq |w|qlq X ejde-39 175 22 ≥ ≥ NOUN ejde-39 175 23 qe qe PROPN ejde-39 175 24 k k PROPN ejde-39 175 25 λ λ PROPN ejde-39 176 1 − − PROPN ejde-39 176 2 e e X ejde-39 176 3 sqq sqq PROPN ejde-39 176 4 ( ( PUNCT ejde-39 176 5 rkλ)q rkλ)q ADV ejde-39 176 6 = = NUM ejde-39 176 7 : : PUNCT ejde-39 176 8 δe δe X ejde-39 176 9 > > X ejde-39 176 10 0 0 PROPN ejde-39 176 11 , , PUNCT ejde-39 176 12 which which PRON ejde-39 176 13 implies imply VERB ejde-39 176 14 the the DET ejde-39 176 15 proof proof NOUN ejde-39 176 16 , , PUNCT ejde-39 176 17 since since SCONJ ejde-39 176 18 reρ reρ NOUN ejde-39 176 19 ( ( PUNCT ejde-39 176 20 w w NOUN ejde-39 176 21 ) ) PUNCT ejde-39 176 22 = = PUNCT ejde-39 176 23 re(w re(w NUM ejde-39 176 24 ) ) PUNCT ejde-39 176 25 , , PUNCT ejde-39 176 26 w w PROPN ejde-39 176 27 ∈ ∈ PROPN ejde-39 176 28 s+ s+ PUNCT ejde-39 176 29 rkλ rkλ NOUN ejde-39 176 30 if if SCONJ ejde-39 176 31 ρ ρ PROPN ejde-39 176 32 ∈ ∈ PROPN ejde-39 176 33 ( ( PUNCT ejde-39 176 34 0 0 NUM ejde-39 176 35 , , PUNCT ejde-39 176 36 rkλ rkλ NOUN ejde-39 176 37 ) ) PUNCT ejde-39 176 38 . . PUNCT ejde-39 177 1 � � PROPN ejde-39 177 2 proposition proposition NOUN ejde-39 177 3 3.2 3.2 NUM ejde-39 177 4 . . PUNCT ejde-39 178 1 for for ADP ejde-39 178 2 any any DET ejde-39 178 3 u u PROPN ejde-39 178 4 ∈ ∈ PROPN ejde-39 178 5 w w NOUN ejde-39 178 6 \ \ PROPN ejde-39 178 7 0 0 PUNCT ejde-39 179 1 and and CCONJ ejde-39 179 2 r r X ejde-39 179 3 > > X ejde-39 179 4 0 0 NUM ejde-39 179 5 , , PUNCT ejde-39 179 6 it it PRON ejde-39 179 7 holds hold VERB ejde-39 179 8 re(tu re(tu ADJ ejde-39 179 9 + + CCONJ ejde-39 179 10 v v NOUN ejde-39 179 11 ) ) PUNCT ejde-39 179 12 → → PUNCT ejde-39 180 1 −∞ −∞ X ejde-39 180 2 as as SCONJ ejde-39 180 3 t→ t→ PRON ejde-39 180 4 + + NOUN ejde-39 180 5 ∞ ∞ NOUN ejde-39 180 6 uniformly uniformly ADV ejde-39 180 7 for for ADP ejde-39 180 8 v v NOUN ejde-39 180 9 ∈ ∈ PROPN ejde-39 180 10 br br NOUN ejde-39 180 11 . . PUNCT ejde-39 181 1 proof proof NOUN ejde-39 181 2 . . PUNCT ejde-39 182 1 observe observe VERB ejde-39 182 2 that that SCONJ ejde-39 182 3 ( ( PUNCT ejde-39 182 4 a2 a2 PROPN ejde-39 182 5 ) ) PUNCT ejde-39 182 6 implies imply VERB ejde-39 182 7 u|u|α u|u|α ADP ejde-39 182 8 d d PROPN ejde-39 182 9 du du X ejde-39 182 10 ( ( PUNCT ejde-39 182 11 |u|−αg(x |u|−αg(x X ejde-39 182 12 , , PUNCT ejde-39 182 13 u u NOUN ejde-39 182 14 ) ) PUNCT ejde-39 182 15 ) ) PUNCT ejde-39 182 16 ≥ ≥ NOUN ejde-39 182 17 0 0 NUM ejde-39 182 18 , , PUNCT ejde-39 182 19 for for ADP ejde-39 182 20 |u| |u| PROPN ejde-39 182 21 ≥ ≥ NOUN ejde-39 182 22 r0 r0 NOUN ejde-39 182 23 . . PUNCT ejde-39 183 1 integrating integrate VERB ejde-39 183 2 this this DET ejde-39 183 3 yields yield NOUN ejde-39 183 4 g(x g(x PROPN ejde-39 183 5 , , PUNCT ejde-39 183 6 u u NOUN ejde-39 183 7 ) ) PUNCT ejde-39 183 8 ≥ ≥ PROPN ejde-39 183 9 c(x)|u|α c(x)|u|α PROPN ejde-39 183 10 > > X ejde-39 183 11 0 0 PUNCT ejde-39 184 1 a.e a.e PROPN ejde-39 184 2 . . PROPN ejde-39 184 3 ω ω PROPN ejde-39 184 4 , , PUNCT ejde-39 184 5 for for ADP ejde-39 184 6 |u| |u| PROPN ejde-39 184 7 ≥ ≥ NOUN ejde-39 184 8 r0 r0 NOUN ejde-39 184 9 , , PUNCT ejde-39 184 10 with with ADP ejde-39 184 11 some some DET ejde-39 184 12 lesbegue lesbegue NOUN ejde-39 184 13 - - PUNCT ejde-39 184 14 measurable measurable NOUN ejde-39 184 15 function function NOUN ejde-39 184 16 c(x c(x NOUN ejde-39 184 17 ) ) PUNCT ejde-39 184 18 ≥ ≥ NOUN ejde-39 184 19 0 0 NUM ejde-39 184 20 . . PUNCT ejde-39 185 1 since since SCONJ ejde-39 185 2 ( ( PUNCT ejde-39 185 3 a2 a2 PROPN ejde-39 185 4 ) ) PUNCT ejde-39 185 5 , , PUNCT ejde-39 185 6 c(x c(x NOUN ejde-39 185 7 ) ) PUNCT ejde-39 185 8 ∈ ∈ NOUN ejde-39 185 9 l∞(ω l∞(ω NOUN ejde-39 185 10 ) ) PUNCT ejde-39 185 11 and and CCONJ ejde-39 185 12 g(x g(x PROPN ejde-39 185 13 , , PUNCT ejde-39 185 14 s s NOUN ejde-39 185 15 ) ) PUNCT ejde-39 185 16 ≥ ≥ NOUN ejde-39 185 17 c(x)|s|α c(x)|s|α PROPN ejde-39 185 18 − − PROPN ejde-39 185 19 c0 c0 NOUN ejde-39 185 20 , , PUNCT ejde-39 185 21 for for ADP ejde-39 185 22 all all DET ejde-39 185 23 s s PART ejde-39 185 24 ∈ ∈ PROPN ejde-39 185 25 r r NOUN ejde-39 185 26 , , PUNCT ejde-39 185 27 a.e a.e PROPN ejde-39 185 28 . . PROPN ejde-39 185 29 ω ω PROPN ejde-39 185 30 with with ADP ejde-39 185 31 some some DET ejde-39 185 32 constant constant ADJ ejde-39 185 33 c0 c0 PROPN ejde-39 185 34 ∈ ∈ PROPN ejde-39 185 35 r. r. PROPN ejde-39 185 36 note note VERB ejde-39 185 37 that that SCONJ ejde-39 185 38 ( ( PUNCT ejde-39 185 39 a2 a2 PROPN ejde-39 185 40 ) ) PUNCT ejde-39 185 41 implies imply VERB ejde-39 185 42 α α X ejde-39 185 43 < < X ejde-39 185 44 γ γ X ejde-39 185 45 < < X ejde-39 185 46 2∗. 2∗. NUM ejde-39 185 47 observe observe VERB ejde-39 185 48 that that SCONJ ejde-39 185 49 c1 c1 PROPN ejde-39 185 50 / / SYM ejde-39 185 51 α(x α(x PROPN ejde-39 185 52 ) ) PUNCT ejde-39 185 53 ( ( PUNCT ejde-39 185 54 u(x u(x PROPN ejde-39 185 55 ) ) PUNCT ejde-39 185 56 + + NUM ejde-39 185 57 v(x)/t v(x)/t NOUN ejde-39 185 58 ) ) PUNCT ejde-39 185 59 → → SYM ejde-39 185 60 c1 c1 NOUN ejde-39 185 61 / / SYM ejde-39 185 62 α(x)u(x α(x)u(x NOUN ejde-39 185 63 ) ) PUNCT ejde-39 185 64 in in ADP ejde-39 185 65 lα(ω lα(ω ADJ ejde-39 185 66 ) ) PUNCT ejde-39 185 67 uniformly uniformly ADV ejde-39 185 68 in in ADP ejde-39 185 69 v v NUM ejde-39 185 70 ∈ ∈ NOUN ejde-39 185 71 br br NOUN ejde-39 185 72 as as SCONJ ejde-39 185 73 t→ t→ PRON ejde-39 185 74 + + PROPN ejde-39 185 75 ∞. ∞. PROPN ejde-39 185 76 indeed indeed ADV ejde-39 185 77 , , PUNCT ejde-39 185 78 using use VERB ejde-39 185 79 the the DET ejde-39 185 80 sobolev sobolev NOUN ejde-39 185 81 inequality inequality NOUN ejde-39 185 82 we we PRON ejde-39 185 83 have∣∣∣ have∣∣∣ ADJ ejde-39 185 84 ∫ ∫ PROPN ejde-39 185 85 c(x c(x NOUN ejde-39 185 86 ) ) PUNCT ejde-39 185 87 ∣∣u+ ∣∣u+ NOUN ejde-39 185 88 v v ADP ejde-39 185 89 t t PROPN ejde-39 185 90 ∣∣α ∣∣α PROPN ejde-39 185 91 dx− dx− NUM ejde-39 185 92 ∫ ∫ PROPN ejde-39 185 93 c(x)|u|α c(x)|u|α PROPN ejde-39 185 94 dx dx PROPN ejde-39 185 95 ∣∣∣ ∣∣∣ PROPN ejde-39 185 96 ≤ ≤ ADV ejde-39 186 1 1 1 NUM ejde-39 186 2 t t NOUN ejde-39 186 3 ∫ ∫ PROPN ejde-39 186 4 c(x)|v|α c(x)|v|α PROPN ejde-39 186 5 dx dx PROPN ejde-39 186 6 ≤ ≤ PROPN ejde-39 186 7 1 1 NUM ejde-39 186 8 t t NOUN ejde-39 186 9 crα crα PROPN ejde-39 186 10 , , PUNCT ejde-39 186 11 v v NOUN ejde-39 186 12 ∈ ∈ NOUN ejde-39 186 13 br br NOUN ejde-39 186 14 for for ADP ejde-39 186 15 some some DET ejde-39 186 16 constant constant ADJ ejde-39 186 17 c c NOUN ejde-39 186 18 which which PRON ejde-39 186 19 does do AUX ejde-39 186 20 not not PART ejde-39 186 21 depend depend VERB ejde-39 186 22 on on ADP ejde-39 186 23 v v NUM ejde-39 186 24 ∈ ∈ PROPN ejde-39 186 25 br br NOUN ejde-39 186 26 . . PUNCT ejde-39 187 1 thus thus ADV ejde-39 187 2 , , PUNCT ejde-39 187 3 uniformly uniformly ADV ejde-39 187 4 in in ADP ejde-39 187 5 v v NUM ejde-39 187 6 ∈ ∈ PROPN ejde-39 187 7 br br NOUN ejde-39 187 8 , , PUNCT ejde-39 187 9 lim lim PROPN ejde-39 187 10 t→∞ t→∞ ADP ejde-39 187 11 1 1 NUM ejde-39 187 12 tα tα PROPN ejde-39 187 13 ∫ ∫ PROPN ejde-39 187 14 g(x g(x NOUN ejde-39 187 15 , , PUNCT ejde-39 187 16 tu+ tu+ NOUN ejde-39 187 17 v v NOUN ejde-39 187 18 ) ) PUNCT ejde-39 187 19 dx dx PROPN ejde-39 187 20 ≥ ≥ PROPN ejde-39 187 21 lim lim PROPN ejde-39 187 22 t→∞ t→∞ PROPN ejde-39 187 23 ( ( PUNCT ejde-39 187 24 ∫ ∫ PROPN ejde-39 187 25 c(x)|u+ c(x)|u+ PROPN ejde-39 187 26 v v ADP ejde-39 187 27 t t PROPN ejde-39 187 28 |αdx− |αdx− NOUN ejde-39 187 29 c0|ω| c0|ω| PROPN ejde-39 187 30 tα tα PROPN ejde-39 187 31 ) ) PUNCT ejde-39 188 1 = = SYM ejde-39 188 2 ∫ ∫ PROPN ejde-39 188 3 c(x)|u|αdx c(x)|u|αdx PROPN ejde-39 188 4 . . PUNCT ejde-39 189 1 this this PRON ejde-39 189 2 implies imply VERB ejde-39 189 3 that that SCONJ ejde-39 189 4 lim lim PROPN ejde-39 189 5 t→∞ t→∞ ADP ejde-39 189 6 1 1 NUM ejde-39 189 7 tα tα PROPN ejde-39 189 8 ( ( PUNCT ejde-39 189 9 1 1 NUM ejde-39 189 10 2 2 NUM ejde-39 189 11 ‖tu+ ‖tu+ NOUN ejde-39 189 12 v‖2w v‖2w NOUN ejde-39 189 13 − − PROPN ejde-39 189 14 ∫ ∫ PROPN ejde-39 189 15 g(x g(x NOUN ejde-39 189 16 , , PUNCT ejde-39 189 17 tu+ tu+ NOUN ejde-39 189 18 v v NOUN ejde-39 189 19 ) ) PUNCT ejde-39 189 20 dx dx PROPN ejde-39 189 21 ) ) PUNCT ejde-39 189 22 ≤ ≤ ADV ejde-39 190 1 − − ADP ejde-39 190 2 ∫ ∫ PROPN ejde-39 190 3 c(x)|u|αdx c(x)|u|αdx PROPN ejde-39 190 4 < < X ejde-39 190 5 0 0 NUM ejde-39 190 6 , , PUNCT ejde-39 190 7 uniformly uniformly ADV ejde-39 190 8 in in ADP ejde-39 190 9 v v NUM ejde-39 190 10 ∈ ∈ NOUN ejde-39 190 11 br br NOUN ejde-39 190 12 . . PUNCT ejde-39 191 1 since since SCONJ ejde-39 191 2 |u |u ADJ ejde-39 191 3 + + CCONJ ejde-39 191 4 v v NOUN ejde-39 191 5 / / SYM ejde-39 191 6 t|qlq t|qlq ADJ ejde-39 191 7 ≤ ≤ NOUN ejde-39 191 8 c c NOUN ejde-39 191 9 ( ( PUNCT ejde-39 191 10 ‖u‖qq ‖u‖qq NOUN ejde-39 191 11 + + CCONJ ejde-39 191 12 sqqr sqqr ADJ ejde-39 191 13 q q X ejde-39 191 14 / / SYM ejde-39 191 15 tq tq NOUN ejde-39 191 16 ) ) PUNCT ejde-39 191 17 and and CCONJ ejde-39 191 18 α α X ejde-39 191 19 > > X ejde-39 191 20 2 2 NUM ejde-39 191 21 > > X ejde-39 191 22 q q X ejde-39 191 23 , , PUNCT ejde-39 191 24 we we PRON ejde-39 191 25 conclude conclude VERB ejde-39 191 26 that that PRON ejde-39 191 27 uniformly uniformly ADV ejde-39 191 28 for for ADP ejde-39 191 29 v v NOUN ejde-39 191 30 ∈ ∈ NOUN ejde-39 191 31 br br NOUN ejde-39 191 32 it it PRON ejde-39 191 33 holds hold VERB ejde-39 191 34 lim lim PROPN ejde-39 191 35 t→∞ t→∞ ADP ejde-39 192 1 re(tu+ re(tu+ NOUN ejde-39 192 2 v v NOUN ejde-39 192 3 ) ) PUNCT ejde-39 192 4 ≤ ≤ NOUN ejde-39 192 5 lim lim PROPN ejde-39 192 6 t→∞ t→∞ PRON ejde-39 192 7 tα−q tα−q PROPN ejde-39 192 8 |u+ |u+ PROPN ejde-39 192 9 v v ADP ejde-39 192 10 t t NOUN ejde-39 192 11 | | ADV ejde-39 192 12 q q X ejde-39 192 13 lq lq VERB ejde-39 192 14 [ [ X ejde-39 192 15 ‖tu+ ‖tu+ X ejde-39 192 16 v‖2w v‖2w NOUN ejde-39 192 17 2tα 2tα NOUN ejde-39 193 1 − − PROPN ejde-39 193 2 ∫ ∫ PROPN ejde-39 194 1 g(x g(x NOUN ejde-39 194 2 , , PUNCT ejde-39 194 3 tu+ tu+ NOUN ejde-39 194 4 v v NOUN ejde-39 194 5 ) ) PUNCT ejde-39 194 6 tα tα VERB ejde-39 194 7 dx dx PROPN ejde-39 194 8 ] ] PUNCT ejde-39 195 1 = = SYM ejde-39 195 2 −∞. −∞. PROPN ejde-39 195 3 � � PROPN ejde-39 195 4 proposition proposition NOUN ejde-39 195 5 3.3 3.3 NUM ejde-39 195 6 . . PUNCT ejde-39 196 1 assume assume VERB ejde-39 196 2 that that SCONJ ejde-39 196 3 e e PROPN ejde-39 196 4 ∈ ∈ PROPN ejde-39 196 5 ( ( PUNCT ejde-39 196 6 0 0 NUM ejde-39 196 7 , , PUNCT ejde-39 196 8 ekλ ekλ ADJ ejde-39 196 9 ) ) PUNCT ejde-39 196 10 , , PUNCT ejde-39 196 11 0 0 NUM ejde-39 196 12 < < X ejde-39 196 13 ρ ρ X ejde-39 196 14 < < X ejde-39 196 15 rkλ rkλ NOUN ejde-39 196 16 . . PUNCT ejde-39 197 1 the the DET ejde-39 197 2 functional functional ADJ ejde-39 197 3 reρ reρ NOUN ejde-39 197 4 satisfies satisfy VERB ejde-39 197 5 the the DET ejde-39 197 6 ( ( PUNCT ejde-39 197 7 ce ce PROPN ejde-39 197 8 ) ) PUNCT ejde-39 197 9 condition condition NOUN ejde-39 197 10 at at ADP ejde-39 197 11 any any DET ejde-39 197 12 level level NOUN ejde-39 197 13 µ µ X ejde-39 197 14 > > X ejde-39 197 15 0 0 NUM ejde-39 197 16 . . PUNCT ejde-39 198 1 ejde-2023/23 ejde-2023/23 NOUN ejde-39 198 2 prescribed prescribe VERB ejde-39 198 3 energy energy NOUN ejde-39 198 4 saddle saddle NOUN ejde-39 198 5 - - PUNCT ejde-39 198 6 point point NOUN ejde-39 198 7 solutions solution NOUN ejde-39 198 8 7 7 NUM ejde-39 198 9 proof proof NOUN ejde-39 198 10 . . PUNCT ejde-39 199 1 by by ADP ejde-39 199 2 corollary corollary ADJ ejde-39 199 3 2.3 2.3 NUM ejde-39 199 4 , , PUNCT ejde-39 199 5 it it PRON ejde-39 199 6 is be AUX ejde-39 199 7 sufficient sufficient ADJ ejde-39 199 8 to to PART ejde-39 199 9 prove prove VERB ejde-39 199 10 that that SCONJ ejde-39 199 11 the the DET ejde-39 199 12 functional functional ADJ ejde-39 199 13 re re NOUN ejde-39 199 14 satisfies satisfie NOUN ejde-39 199 15 the the DET ejde-39 199 16 ( ( PUNCT ejde-39 199 17 ce ce PROPN ejde-39 199 18 ) ) PUNCT ejde-39 199 19 condition condition NOUN ejde-39 199 20 at at ADP ejde-39 199 21 any any DET ejde-39 199 22 µ µ X ejde-39 199 23 > > X ejde-39 199 24 0 0 X ejde-39 199 25 . . PUNCT ejde-39 200 1 assume assume VERB ejde-39 200 2 that that SCONJ ejde-39 200 3 ( ( PUNCT ejde-39 200 4 um um INTJ ejde-39 200 5 ) ) PUNCT ejde-39 200 6 is be AUX ejde-39 200 7 a a DET ejde-39 200 8 ( ( PUNCT ejde-39 200 9 ce ce NOUN ejde-39 200 10 ) ) PUNCT ejde-39 200 11 sequence sequence NOUN ejde-39 200 12 for for ADP ejde-39 200 13 re re PROPN ejde-39 200 14 , , PUNCT ejde-39 200 15 i.e. i.e. X ejde-39 200 16 , , PUNCT ejde-39 200 17 µm µm ADP ejde-39 200 18 : : PUNCT ejde-39 200 19 = = SYM ejde-39 200 20 re(um)→ re(um)→ PROPN ejde-39 200 21 µ µ X ejde-39 200 22 > > X ejde-39 200 23 0 0 PUNCT ejde-39 200 24 and and CCONJ ejde-39 200 25 ‖dre(um)‖∗(1 ‖dre(um)‖∗(1 PROPN ejde-39 200 26 + + NUM ejde-39 200 27 ‖um‖1)→ ‖um‖1)→ NOUN ejde-39 200 28 0 0 PUNCT ejde-39 200 29 as as ADP ejde-39 200 30 m→ m→ PROPN ejde-39 200 31 + + NOUN ejde-39 200 32 ∞. ∞. PROPN ejde-39 200 33 then then ADV ejde-39 200 34 αµm αµm PROPN ejde-39 201 1 + + CCONJ ejde-39 201 2 o(1)‖um‖1(1 o(1)‖um‖1(1 ADJ ejde-39 201 3 + + ADJ ejde-39 201 4 ‖um‖1)−1 ‖um‖1)−1 ADP ejde-39 201 5 = = NOUN ejde-39 201 6 αre(um)−dre(um)(um αre(um)−dre(um)(um NOUN ejde-39 201 7 ) ) PUNCT ejde-39 201 8 = = PUNCT ejde-39 202 1 q q NOUN ejde-39 202 2 |um|qlq |um|qlq ADJ ejde-39 202 3 ( ( PUNCT ejde-39 202 4 α− α− ADP ejde-39 202 5 2 2 NUM ejde-39 202 6 2 2 NUM ejde-39 202 7 hλ(um hλ(um NOUN ejde-39 202 8 ) ) PUNCT ejde-39 203 1 + + NUM ejde-39 203 2 ∫ ∫ PROPN ejde-39 203 3 ( ( PUNCT ejde-39 203 4 g(x g(x NOUN ejde-39 203 5 , , PUNCT ejde-39 203 6 um)um um)um ADP ejde-39 203 7 − − NOUN ejde-39 203 8 αg(x αg(x X ejde-39 203 9 , , PUNCT ejde-39 203 10 um um INTJ ejde-39 203 11 ) ) PUNCT ejde-39 203 12 ) ) PUNCT ejde-39 203 13 dx+ dx+ NOUN ejde-39 204 1 µm|um|qlq µm|um|qlq PROPN ejde-39 204 2 − − PROPN ejde-39 204 3 qe qe PROPN ejde-39 204 4 ) ) PUNCT ejde-39 204 5 ≥ ≥ PROPN ejde-39 204 6 q q X ejde-39 204 7 |um|qlq |um|qlq ADJ ejde-39 204 8 ( ( PUNCT ejde-39 204 9 α− α− ADP ejde-39 204 10 2 2 NUM ejde-39 204 11 2 2 NUM ejde-39 204 12 hλ(um hλ(um NOUN ejde-39 204 13 ) ) PUNCT ejde-39 205 1 + + CCONJ ejde-39 205 2 ln(ω)essinfx∈ω ln(ω)essinfx∈ω ADJ ejde-39 205 3 , , PUNCT ejde-39 205 4 s∈r s∈r NOUN ejde-39 205 5 ( ( PUNCT ejde-39 205 6 g(x g(x PROPN ejde-39 205 7 , , PUNCT ejde-39 205 8 s)s− s)s− NOUN ejde-39 205 9 αg(x αg(x NUM ejde-39 205 10 , , PUNCT ejde-39 205 11 s))− s))− ADJ ejde-39 205 12 qe qe NOUN ejde-39 205 13 ) ) PUNCT ejde-39 205 14 by by ADP ejde-39 205 15 ( ( PUNCT ejde-39 205 16 a2 a2 PROPN ejde-39 205 17 ) ) PUNCT ejde-39 205 18 , , PUNCT ejde-39 205 19 ess ess PROPN ejde-39 205 20 infx∈ω infx∈ω PROPN ejde-39 205 21 , , PUNCT ejde-39 205 22 s∈r s∈r NOUN ejde-39 205 23 ( ( PUNCT ejde-39 205 24 g(x g(x PROPN ejde-39 205 25 , , PUNCT ejde-39 205 26 s)s−αg(x s)s−αg(x ADJ ejde-39 205 27 , , PUNCT ejde-39 205 28 s s NOUN ejde-39 205 29 ) ) PUNCT ejde-39 205 30 ) ) PUNCT ejde-39 206 1 = = PRON ejde-39 206 2 : : PUNCT ejde-39 206 3 c0 c0 X ejde-39 206 4 > > X ejde-39 206 5 −∞. −∞. PROPN ejde-39 206 6 hence hence ADV ejde-39 206 7 hλ(um hλ(um PROPN ejde-39 206 8 ) ) PUNCT ejde-39 206 9 ≤ ≤ PROPN ejde-39 206 10 c1 c1 NOUN ejde-39 206 11 ( ( PUNCT ejde-39 206 12 1 1 NUM ejde-39 206 13 + + CCONJ ejde-39 206 14 |um|qlq |um|qlq ADJ ejde-39 206 15 ) ) PUNCT ejde-39 206 16 , , PUNCT ejde-39 206 17 where where SCONJ ejde-39 206 18 0 0 X ejde-39 206 19 < < X ejde-39 206 20 c1 c1 PROPN ejde-39 206 21 < < X ejde-39 206 22 + + NOUN ejde-39 206 23 ∞ ∞ PROPN ejde-39 206 24 does do AUX ejde-39 206 25 not not PART ejde-39 206 26 depend depend VERB ejde-39 206 27 on on ADP ejde-39 206 28 m m PROPN ejde-39 206 29 = = SYM ejde-39 206 30 1 1 NUM ejde-39 206 31 , , PUNCT ejde-39 206 32 2 2 NUM ejde-39 206 33 , , PUNCT ejde-39 206 34 . . PUNCT ejde-39 206 35 . . PUNCT ejde-39 207 1 . . PUNCT ejde-39 208 1 , , PUNCT ejde-39 208 2 and and CCONJ ejde-39 208 3 therefore therefore ADV ejde-39 208 4 ‖um‖2w ‖um‖2w ADJ ejde-39 208 5 ≤ ≤ NOUN ejde-39 209 1 λ|um|2l2 λ|um|2l2 PROPN ejde-39 209 2 + + NUM ejde-39 209 3 c1 c1 PROPN ejde-39 209 4 ( ( PUNCT ejde-39 209 5 1 1 NUM ejde-39 209 6 + + CCONJ ejde-39 209 7 |um|qlq |um|qlq ADJ ejde-39 209 8 ) ) PUNCT ejde-39 209 9 , , PUNCT ejde-39 209 10 m m VERB ejde-39 209 11 = = NOUN ejde-39 209 12 1 1 NUM ejde-39 209 13 , , PUNCT ejde-39 209 14 2 2 NUM ejde-39 209 15 , , PUNCT ejde-39 209 16 . . PUNCT ejde-39 209 17 . . PUNCT ejde-39 209 18 . . PUNCT ejde-39 209 19 . . PUNCT ejde-39 210 1 ( ( PUNCT ejde-39 210 2 3.2 3.2 NUM ejde-39 210 3 ) ) PUNCT ejde-39 210 4 thus thus ADV ejde-39 210 5 , , PUNCT ejde-39 210 6 if if SCONJ ejde-39 210 7 |um|l2 |um|l2 NOUN ejde-39 210 8 is be AUX ejde-39 210 9 bounded bound VERB ejde-39 210 10 , , PUNCT ejde-39 210 11 then then ADV ejde-39 210 12 ‖um‖w ‖um‖w PROPN ejde-39 210 13 is be AUX ejde-39 210 14 also also ADV ejde-39 210 15 bounded bound VERB ejde-39 210 16 . . PUNCT ejde-39 211 1 if if SCONJ ejde-39 211 2 |umj |umj X ejde-39 211 3 |l2 |l2 NOUN ejde-39 211 4 → → SYM ejde-39 211 5 ∞ ∞ PROPN ejde-39 211 6 , , PUNCT ejde-39 211 7 for for ADP ejde-39 211 8 some some DET ejde-39 211 9 subsequence subsequence NOUN ejde-39 211 10 ( ( PUNCT ejde-39 211 11 mj mj NOUN ejde-39 211 12 ) ) PUNCT ejde-39 211 13 such such ADJ ejde-39 211 14 that that SCONJ ejde-39 211 15 mj mj PROPN ejde-39 211 16 → → PUNCT ejde-39 211 17 + + ADJ ejde-39 211 18 ∞ ∞ PROPN ejde-39 211 19 as as ADP ejde-39 211 20 j j PROPN ejde-39 211 21 → → SYM ejde-39 211 22 + + PROPN ejde-39 211 23 ∞ ∞ PROPN ejde-39 211 24 , , PUNCT ejde-39 211 25 then then ADV ejde-39 211 26 by by ADP ejde-39 211 27 ( ( PUNCT ejde-39 211 28 3.2 3.2 NUM ejde-39 211 29 ) ) PUNCT ejde-39 211 30 limj→∞ limj→∞ PROPN ejde-39 211 31 hλ(umj hλ(umj PROPN ejde-39 211 32 ) ) PUNCT ejde-39 211 33 |umj |umj NOUN ejde-39 211 34 | | ADV ejde-39 211 35 2 2 NUM ejde-39 211 36 l2 l2 VERB ejde-39 211 37 ≤ ≤ NOUN ejde-39 211 38 0 0 NUM ejde-39 211 39 , , PUNCT ejde-39 211 40 and and CCONJ ejde-39 211 41 consequently consequently ADV ejde-39 211 42 , , PUNCT ejde-39 211 43 we we PRON ejde-39 211 44 obtain obtain VERB ejde-39 211 45 a a DET ejde-39 211 46 contradiction contradiction NOUN ejde-39 211 47 : : PUNCT ejde-39 211 48 0 0 NUM ejde-39 211 49 < < X ejde-39 211 50 µ µ X ejde-39 211 51 = = SYM ejde-39 211 52 lim lim PROPN ejde-39 211 53 j→∞ j→∞ NOUN ejde-39 211 54 re(umj re(umj PROPN ejde-39 211 55 ) ) PUNCT ejde-39 212 1 = = SYM ejde-39 212 2 lim lim PROPN ejde-39 212 3 j→∞ j→∞ NOUN ejde-39 212 4 q q PROPN ejde-39 212 5 |umj |umj NOUN ejde-39 212 6 |2l2 |2l2 PRON ejde-39 212 7 |umj |umj VERB ejde-39 213 1 | | ADV ejde-39 213 2 q q X ejde-39 213 3 lq lq ADP ejde-39 213 4 [ [ X ejde-39 213 5 1 1 NUM ejde-39 213 6 2 2 NUM ejde-39 213 7 hλ(umj hλ(umj NOUN ejde-39 213 8 ) ) PUNCT ejde-39 213 9 |umj |umj NOUN ejde-39 214 1 |2l2 |2l2 PRON ejde-39 214 2 − − PROPN ejde-39 214 3 ∫ ∫ PROPN ejde-39 214 4 g(x g(x NOUN ejde-39 214 5 , , PUNCT ejde-39 214 6 umj umj ADJ ejde-39 214 7 ) ) PUNCT ejde-39 214 8 |umj |umj NOUN ejde-39 214 9 |2l2 |2l2 PRON ejde-39 214 10 dx− dx− NUM ejde-39 214 11 e e X ejde-39 214 12 |umj |umj X ejde-39 214 13 |2l2 |2l2 X ejde-39 214 14 ] ] PUNCT ejde-39 214 15 ≤ ≤ NUM ejde-39 214 16 0 0 NUM ejde-39 214 17 . . PUNCT ejde-39 215 1 thus thus ADV ejde-39 215 2 , , PUNCT ejde-39 215 3 ( ( PUNCT ejde-39 215 4 um um INTJ ejde-39 215 5 ) ) PUNCT ejde-39 215 6 is be AUX ejde-39 215 7 bounded bound VERB ejde-39 215 8 and and CCONJ ejde-39 215 9 we we PRON ejde-39 215 10 may may AUX ejde-39 215 11 assume assume VERB ejde-39 215 12 that that SCONJ ejde-39 215 13 um um INTJ ejde-39 215 14 ⇀ ⇀ NUM ejde-39 215 15 u u NOUN ejde-39 215 16 weakly weakly ADV ejde-39 215 17 in in ADP ejde-39 215 18 w w PROPN ejde-39 215 19 and and CCONJ ejde-39 215 20 um um INTJ ejde-39 215 21 → → SYM ejde-39 215 22 u u NOUN ejde-39 215 23 strongly strongly ADV ejde-39 215 24 in in ADP ejde-39 215 25 lr(ω lr(ω NOUN ejde-39 215 26 ) ) PUNCT ejde-39 215 27 , , PUNCT ejde-39 215 28 r r NOUN ejde-39 215 29 ∈ ∈ PROPN ejde-39 216 1 [ [ X ejde-39 216 2 1 1 NUM ejde-39 216 3 , , PUNCT ejde-39 216 4 2∗ 2∗ NUM ejde-39 216 5 ) ) PUNCT ejde-39 216 6 , , PUNCT ejde-39 216 7 as as ADP ejde-39 216 8 m→∞. m→∞. PROPN ejde-39 216 9 in in ADP ejde-39 216 10 particular particular ADJ ejde-39 216 11 , , PUNCT ejde-39 216 12 this this PRON ejde-39 216 13 gives∫ gives∫ NOUN ejde-39 216 14 g(x g(x NOUN ejde-39 216 15 , , PUNCT ejde-39 216 16 um)dx→ um)dx→ PROPN ejde-39 216 17 ∫ ∫ NOUN ejde-39 216 18 g(x g(x PROPN ejde-39 216 19 , , PUNCT ejde-39 216 20 u)dx u)dx PROPN ejde-39 216 21 , , PUNCT ejde-39 216 22 ‖um‖qq ‖um‖qq PROPN ejde-39 216 23 → → SYM ejde-39 216 24 ‖u‖qq ‖u‖qq NOUN ejde-39 216 25 as as ADP ejde-39 216 26 m→ m→ PROPN ejde-39 216 27 + + NOUN ejde-39 216 28 ∞. ∞. PROPN ejde-39 216 29 ( ( PUNCT ejde-39 216 30 3.3 3.3 NUM ejde-39 216 31 ) ) PUNCT ejde-39 216 32 by by ADP ejde-39 216 33 the the DET ejde-39 216 34 convergence convergence NOUN ejde-39 216 35 ‖dre(um)‖∗ ‖dre(um)‖∗ NOUN ejde-39 216 36 → → SYM ejde-39 216 37 0 0 NUM ejde-39 216 38 we we PRON ejde-39 216 39 obtain obtain AUX ejde-39 216 40 dre(um)(u− dre(um)(u− ADJ ejde-39 216 41 um)→ um)→ SYM ejde-39 216 42 0 0 NUM ejde-39 216 43 as as ADP ejde-39 216 44 m→ m→ PROPN ejde-39 216 45 + + NOUN ejde-39 216 46 ∞. ∞. PROPN ejde-39 216 47 hence hence ADV ejde-39 216 48 by by ADP ejde-39 216 49 ( ( PUNCT ejde-39 216 50 3.3 3.3 NUM ejde-39 216 51 ) ) PUNCT ejde-39 216 52 , , PUNCT ejde-39 216 53 we we PRON ejde-39 216 54 obtain obtain VERB ejde-39 216 55 that that SCONJ ejde-39 216 56 〈 〈 PROPN ejde-39 216 57 −∆um −∆um PROPN ejde-39 216 58 , , PUNCT ejde-39 216 59 u−um u−um PROPN ejde-39 216 60 〉 〉 PROPN ejde-39 216 61 → → SYM ejde-39 216 62 0 0 NUM ejde-39 216 63 . . PUNCT ejde-39 216 64 thus thus ADV ejde-39 216 65 by by ADP ejde-39 216 66 the the DET ejde-39 216 67 s+ s+ ADJ ejde-39 216 68 property property NOUN ejde-39 216 69 of of ADP ejde-39 216 70 the the DET ejde-39 216 71 laplace laplace NOUN ejde-39 216 72 operator operator NOUN ejde-39 216 73 ( ( PUNCT ejde-39 216 74 see see VERB ejde-39 216 75 [ [ X ejde-39 216 76 13 13 NUM ejde-39 216 77 ] ] SYM ejde-39 216 78 ) ) PUNCT ejde-39 216 79 we we PRON ejde-39 216 80 derive derive VERB ejde-39 216 81 that that SCONJ ejde-39 216 82 um um INTJ ejde-39 216 83 → → SYM ejde-39 216 84 u u NOUN ejde-39 216 85 strongly strongly ADV ejde-39 216 86 in in ADP ejde-39 216 87 w w PROPN ejde-39 216 88 1,2(ω 1,2(ω NUM ejde-39 216 89 ) ) PUNCT ejde-39 216 90 . . PUNCT ejde-39 217 1 � � PROPN ejde-39 217 2 remark remark VERB ejde-39 217 3 3.4 3.4 NUM ejde-39 217 4 . . PUNCT ejde-39 218 1 since since SCONJ ejde-39 218 2 ‖um‖1 ‖um‖1 PROPN ejde-39 218 3 > > X ejde-39 218 4 ρ ρ PROPN ejde-39 218 5 , , PUNCT ejde-39 218 6 for for ADP ejde-39 218 7 all all DET ejde-39 218 8 m m NOUN ejde-39 218 9 and and CCONJ ejde-39 218 10 reρ reρ X ejde-39 218 11 ( ( PUNCT ejde-39 218 12 um um INTJ ejde-39 218 13 ) ) PUNCT ejde-39 218 14 → → SYM ejde-39 218 15 µ µ X ejde-39 218 16 ∈ ∈ NOUN ejde-39 218 17 ( ( PUNCT ejde-39 218 18 0,+∞ 0,+∞ NUM ejde-39 218 19 ) ) PUNCT ejde-39 218 20 , , PUNCT ejde-39 218 21 it it PRON ejde-39 218 22 follows follow VERB ejde-39 218 23 that that SCONJ ejde-39 218 24 |um|lq |um|lq PUNCT ejde-39 218 25 dos dos AUX ejde-39 218 26 not not PART ejde-39 218 27 approach approach VERB ejde-39 218 28 0 0 NUM ejde-39 218 29 . . PUNCT ejde-39 219 1 4 4 NUM ejde-39 219 2 . . X ejde-39 219 3 proof proof NOUN ejde-39 219 4 of of ADP ejde-39 219 5 theorem theorem ADJ ejde-39 219 6 1.1 1.1 NUM ejde-39 219 7 let let VERB ejde-39 219 8 λ λ X ejde-39 219 9 ∈ ∈ PROPN ejde-39 219 10 ( ( PUNCT ejde-39 219 11 λk λk X ejde-39 219 12 , , PUNCT ejde-39 219 13 λk+1 λk+1 X ejde-39 219 14 ) ) PUNCT ejde-39 219 15 , , PUNCT ejde-39 219 16 e e PROPN ejde-39 219 17 ∈ ∈ PROPN ejde-39 219 18 ( ( PUNCT ejde-39 219 19 0 0 NUM ejde-39 219 20 , , PUNCT ejde-39 219 21 ekλ ekλ ADJ ejde-39 219 22 ) ) PUNCT ejde-39 219 23 and and CCONJ ejde-39 219 24 0 0 NUM ejde-39 219 25 < < X ejde-39 219 26 ρ ρ X ejde-39 219 27 < < X ejde-39 219 28 rkλ rkλ NOUN ejde-39 219 29 . . PUNCT ejde-39 220 1 for for ADP ejde-39 220 2 t t PROPN ejde-39 220 3 > > X ejde-39 220 4 rkλ rkλ PROPN ejde-39 220 5 , , PUNCT ejde-39 220 6 take take VERB ejde-39 220 7 ū+ ū+ NUM ejde-39 220 8 ∈ ∈ NOUN ejde-39 220 9 s+ s+ PUNCT ejde-39 220 10 1 1 NUM ejde-39 220 11 , , PUNCT ejde-39 220 12 and and CCONJ ejde-39 220 13 define define VERB ejde-39 220 14 b0 b0 NOUN ejde-39 220 15 = = SYM ejde-39 220 16 b0(t b0(t PROPN ejde-39 220 17 ) ) PUNCT ejde-39 220 18 : : PUNCT ejde-39 221 1 = = SYM ejde-39 221 2 { { PUNCT ejde-39 221 3 u u NOUN ejde-39 221 4 = = PUNCT ejde-39 221 5 tū+ tū+ PROPN ejde-39 221 6 + + NUM ejde-39 221 7 sv sv NOUN ejde-39 221 8 : : PUNCT ejde-39 221 9 v v X ejde-39 221 10 ∈ ∈ NOUN ejde-39 221 11 s−1 s−1 PROPN ejde-39 221 12 , , PUNCT ejde-39 221 13 ( ( PUNCT ejde-39 221 14 0 0 X ejde-39 221 15 < < X ejde-39 221 16 t t X ejde-39 221 17 < < X ejde-39 221 18 t t PROPN ejde-39 221 19 , , PUNCT ejde-39 221 20 s s PART ejde-39 221 21 = = X ejde-39 221 22 t t PROPN ejde-39 221 23 ) ) PUNCT ejde-39 221 24 or or CCONJ ejde-39 221 25 ( ( PUNCT ejde-39 221 26 t t PROPN ejde-39 221 27 ∈ ∈ PROPN ejde-39 221 28 { { PUNCT ejde-39 221 29 0 0 NUM ejde-39 221 30 , , PUNCT ejde-39 221 31 t t PROPN ejde-39 221 32 } } PUNCT ejde-39 221 33 , , PUNCT ejde-39 221 34 0 0 NUM ejde-39 221 35 ≤ ≤ NUM ejde-39 221 36 s s PART ejde-39 221 37 ≤ ≤ PROPN ejde-39 221 38 t t NOUN ejde-39 221 39 ) ) PUNCT ejde-39 221 40 } } PUNCT ejde-39 221 41 , , PUNCT ejde-39 221 42 b b X ejde-39 221 43 = = SYM ejde-39 221 44 b(t b(t PROPN ejde-39 221 45 ) ) PUNCT ejde-39 221 46 : : PUNCT ejde-39 221 47 = = SYM ejde-39 221 48 { { PUNCT ejde-39 221 49 u u NOUN ejde-39 221 50 = = PUNCT ejde-39 221 51 tū+ tū+ PROPN ejde-39 221 52 + + NUM ejde-39 221 53 sv sv NOUN ejde-39 221 54 : : PUNCT ejde-39 221 55 v v X ejde-39 221 56 ∈ ∈ NOUN ejde-39 221 57 s−1 s−1 PROPN ejde-39 221 58 , , PUNCT ejde-39 221 59 0 0 NUM ejde-39 221 60 < < X ejde-39 221 61 t t X ejde-39 221 62 < < X ejde-39 221 63 t t PROPN ejde-39 221 64 , , PUNCT ejde-39 221 65 0 0 NUM ejde-39 221 66 ≤ ≤ NUM ejde-39 221 67 s s PART ejde-39 221 68 ≤ ≤ NUM ejde-39 221 69 t t PROPN ejde-39 221 70 } } PUNCT ejde-39 221 71 , , PUNCT ejde-39 221 72 bc0 bc0 X ejde-39 221 73 = = PROPN ejde-39 221 74 bc0(t bc0(t PROPN ejde-39 221 75 ) ) PUNCT ejde-39 221 76 : : PUNCT ejde-39 222 1 = = SYM ejde-39 222 2 { { PUNCT ejde-39 222 3 u u NOUN ejde-39 222 4 = = PUNCT ejde-39 222 5 tū+ tū+ PROPN ejde-39 222 6 + + NUM ejde-39 222 7 sv sv NOUN ejde-39 222 8 : : PUNCT ejde-39 222 9 v v X ejde-39 222 10 ∈ ∈ NOUN ejde-39 222 11 s−1 s−1 PROPN ejde-39 222 12 , , PUNCT ejde-39 222 13 ( ( PUNCT ejde-39 222 14 0 0 X ejde-39 222 15 < < X ejde-39 222 16 t t X ejde-39 222 17 < < X ejde-39 222 18 t t PROPN ejde-39 222 19 , , PUNCT ejde-39 222 20 s s PART ejde-39 222 21 = = PROPN ejde-39 222 22 t t PROPN ejde-39 222 23 ) ) PUNCT ejde-39 222 24 } } PUNCT ejde-39 222 25 , , PUNCT ejde-39 222 26 bd0 bd0 X ejde-39 222 27 = = SYM ejde-39 222 28 bd0 bd0 X ejde-39 222 29 ( ( PUNCT ejde-39 222 30 t t PROPN ejde-39 222 31 ) ) PUNCT ejde-39 222 32 : : PUNCT ejde-39 222 33 = = SYM ejde-39 222 34 { { PUNCT ejde-39 222 35 u u NOUN ejde-39 222 36 = = PUNCT ejde-39 222 37 tū+ tū+ PROPN ejde-39 222 38 + + NUM ejde-39 222 39 sv sv NOUN ejde-39 222 40 : : PUNCT ejde-39 222 41 v v X ejde-39 222 42 ∈ ∈ NOUN ejde-39 222 43 s−1 s−1 PROPN ejde-39 222 44 , , PUNCT ejde-39 222 45 t t PROPN ejde-39 222 46 ∈ ∈ PROPN ejde-39 222 47 { { PUNCT ejde-39 222 48 0 0 NUM ejde-39 222 49 , , PUNCT ejde-39 222 50 t t PROPN ejde-39 222 51 } } PUNCT ejde-39 222 52 , , PUNCT ejde-39 222 53 0 0 NUM ejde-39 222 54 ≤ ≤ NUM ejde-39 222 55 s s PART ejde-39 222 56 ≤ ≤ NUM ejde-39 222 57 t t PROPN ejde-39 222 58 } } PUNCT ejde-39 222 59 . . PUNCT ejde-39 223 1 8 8 NUM ejde-39 223 2 y. y. PROPN ejde-39 223 3 il’yasov il’yasov PROPN ejde-39 223 4 , , PUNCT ejde-39 223 5 e. e. PROPN ejde-39 223 6 d. d. PROPN ejde-39 223 7 silva silva PROPN ejde-39 223 8 , , PUNCT ejde-39 223 9 m. m. PROPN ejde-39 223 10 l. l. PROPN ejde-39 223 11 silva silva PROPN ejde-39 223 12 ejde-2023/23 ejde-2023/23 PROPN ejde-39 223 13 observe observe VERB ejde-39 223 14 that that SCONJ ejde-39 223 15 if if SCONJ ejde-39 223 16 u u PRON ejde-39 223 17 = = PUNCT ejde-39 223 18 tū+ tū+ PUNCT ejde-39 223 19 + + NOUN ejde-39 223 20 tv tv NOUN ejde-39 223 21 , , PUNCT ejde-39 223 22 u+ u+ NOUN ejde-39 223 23 ∈ ∈ PROPN ejde-39 223 24 s+ s+ PUNCT ejde-39 223 25 1 1 NUM ejde-39 223 26 , , PUNCT ejde-39 223 27 v v NOUN ejde-39 223 28 ∈ ∈ NOUN ejde-39 223 29 s−1 s−1 NOUN ejde-39 223 30 , , PUNCT ejde-39 223 31 then then ADV ejde-39 223 32 hλ(u hλ(u NUM ejde-39 223 33 ) ) PUNCT ejde-39 223 34 = = SYM ejde-39 223 35 t2‖ū+‖21−t t2‖ū+‖21−t X ejde-39 223 36 2‖v‖21 2‖v‖21 NUM ejde-39 223 37 = = SYM ejde-39 223 38 ( ( PUNCT ejde-39 223 39 t2 t2 PROPN ejde-39 223 40 − − PROPN ejde-39 223 41 t t PROPN ejde-39 223 42 2 2 NUM ejde-39 223 43 ) ) PUNCT ejde-39 223 44 < < X ejde-39 223 45 0 0 NUM ejde-39 223 46 for for ADP ejde-39 223 47 t t PROPN ejde-39 223 48 > > PUNCT ejde-39 223 49 t. t. PROPN ejde-39 223 50 this this PRON ejde-39 223 51 implies imply VERB ejde-39 223 52 bc(t bc(t X ejde-39 223 53 ) ) PUNCT ejde-39 224 1 : : PUNCT ejde-39 225 1 = = SYM ejde-39 225 2 sup sup X ejde-39 225 3 u∈bc0 u∈bc0 INTJ ejde-39 225 4 reρ reρ X ejde-39 225 5 ( ( PUNCT ejde-39 225 6 u u NOUN ejde-39 225 7 ) ) PUNCT ejde-39 225 8 = = SYM ejde-39 225 9 sup sup NOUN ejde-39 225 10 u∈bc0 u∈bc0 PUNCT ejde-39 225 11 φρ(‖u‖1 φρ(‖u‖1 NOUN ejde-39 225 12 ) ) PUNCT ejde-39 225 13 1 1 NUM ejde-39 225 14 2hλ(u)− 2hλ(u)− NUM ejde-39 225 15 ∫ ∫ NOUN ejde-39 225 16 g(x g(x NOUN ejde-39 225 17 , , PUNCT ejde-39 225 18 u u NOUN ejde-39 225 19 ) ) PUNCT ejde-39 225 20 dx− dx− X ejde-39 225 21 e e PROPN ejde-39 225 22 1 1 NUM ejde-39 225 23 q q NOUN ejde-39 225 24 |u| |u| NOUN ejde-39 225 25 q q X ejde-39 225 26 lq lq VERB ejde-39 225 27 ≤ ≤ NUM ejde-39 225 28 0 0 NUM ejde-39 225 29 , , PUNCT ejde-39 225 30 for for ADP ejde-39 225 31 ρ ρ PROPN ejde-39 225 32 > > X ejde-39 225 33 0 0 PROPN ejde-39 225 34 . . PUNCT ejde-39 225 35 note note VERB ejde-39 225 36 that that SCONJ ejde-39 225 37 hλ(v hλ(v NUM ejde-39 225 38 ) ) PUNCT ejde-39 225 39 < < X ejde-39 225 40 0 0 NUM ejde-39 225 41 , , PUNCT ejde-39 225 42 for for ADP ejde-39 225 43 v v NOUN ejde-39 225 44 ∈ ∈ NOUN ejde-39 225 45 s−1 s−1 PROPN ejde-39 225 46 . . PUNCT ejde-39 226 1 this this PRON ejde-39 226 2 by by ADP ejde-39 226 3 proposition proposition NOUN ejde-39 226 4 3.2 3.2 NUM ejde-39 226 5 implies imply VERB ejde-39 226 6 that that SCONJ ejde-39 226 7 bd(t bd(t PUNCT ejde-39 226 8 ) ) PUNCT ejde-39 226 9 : : PUNCT ejde-39 226 10 = = SYM ejde-39 226 11 sup sup NOUN ejde-39 226 12 u∈bd0 u∈bd0 NOUN ejde-39 226 13 reρ reρ NOUN ejde-39 226 14 ( ( PUNCT ejde-39 226 15 u u NOUN ejde-39 226 16 ) ) PUNCT ejde-39 226 17 ≤ ≤ NOUN ejde-39 226 18 0 0 NUM ejde-39 226 19 , , PUNCT ejde-39 226 20 for for ADP ejde-39 226 21 sufficiently sufficiently ADV ejde-39 226 22 large large ADJ ejde-39 226 23 t t NOUN ejde-39 226 24 . . PUNCT ejde-39 227 1 thus thus ADV ejde-39 227 2 , , PUNCT ejde-39 227 3 by by ADP ejde-39 227 4 proposition proposition NOUN ejde-39 227 5 3.1 3.1 NUM ejde-39 227 6 , , PUNCT ejde-39 227 7 for for ADP ejde-39 227 8 ρ ρ PROPN ejde-39 227 9 ∈ ∈ PROPN ejde-39 227 10 ( ( PUNCT ejde-39 227 11 0 0 NUM ejde-39 227 12 , , PUNCT ejde-39 227 13 rkλ rkλ NOUN ejde-39 227 14 ) ) PUNCT ejde-39 227 15 and and CCONJ ejde-39 227 16 sufficiently sufficiently ADV ejde-39 227 17 large large ADJ ejde-39 227 18 t t PROPN ejde-39 227 19 > > X ejde-39 227 20 rkλ rkλ PROPN ejde-39 227 21 , , PUNCT ejde-39 227 22 it it PRON ejde-39 227 23 holds hold VERB ejde-39 227 24 b b NOUN ejde-39 227 25 : : PUNCT ejde-39 227 26 = = SYM ejde-39 227 27 sup sup NOUN ejde-39 227 28 u∈b0 u∈b0 NOUN ejde-39 227 29 reρ reρ NOUN ejde-39 227 30 ( ( PUNCT ejde-39 227 31 u u NOUN ejde-39 227 32 ) ) PUNCT ejde-39 227 33 ≤ ≤ NOUN ejde-39 227 34 0 0 NUM ejde-39 228 1 < < X ejde-39 228 2 inf inf PROPN ejde-39 228 3 u∈s+ u∈s+ ADJ ejde-39 228 4 rk rk PROPN ejde-39 228 5 λ λ PROPN ejde-39 228 6 reρ reρ X ejde-39 228 7 ( ( PUNCT ejde-39 228 8 u u NOUN ejde-39 228 9 ) ) PUNCT ejde-39 228 10 = = NOUN ejde-39 228 11 : : PUNCT ejde-39 228 12 a. a. NOUN ejde-39 228 13 let let VERB ejde-39 228 14 λ λ X ejde-39 228 15 ∈ ∈ PROPN ejde-39 228 16 ( ( PUNCT ejde-39 228 17 λk λk X ejde-39 228 18 , , PUNCT ejde-39 228 19 λk+1 λk+1 X ejde-39 228 20 ) ) PUNCT ejde-39 228 21 , , PUNCT ejde-39 228 22 e e PROPN ejde-39 228 23 ∈ ∈ PROPN ejde-39 228 24 ( ( PUNCT ejde-39 228 25 0 0 NUM ejde-39 228 26 , , PUNCT ejde-39 228 27 ekλ ekλ ADJ ejde-39 228 28 ) ) PUNCT ejde-39 228 29 and and CCONJ ejde-39 228 30 0 0 NUM ejde-39 228 31 < < X ejde-39 228 32 ρ ρ X ejde-39 228 33 < < X ejde-39 228 34 min{rkλ min{rkλ PROPN ejde-39 228 35 , , PUNCT ejde-39 228 36 ρ(e ρ(e PROPN ejde-39 228 37 ) ) PUNCT ejde-39 228 38 } } PUNCT ejde-39 228 39 . . PUNCT ejde-39 229 1 consider consider VERB ejde-39 229 2 µkλ(e µkλ(e NOUN ejde-39 229 3 ) ) PUNCT ejde-39 229 4 : : PUNCT ejde-39 230 1 = = SYM ejde-39 230 2 inf inf PROPN ejde-39 230 3 h∈γ h∈γ NOUN ejde-39 230 4 max max PROPN ejde-39 230 5 u∈b u∈b PROPN ejde-39 230 6 reρ reρ PROPN ejde-39 230 7 ( ( PUNCT ejde-39 230 8 h(u h(u PROPN ejde-39 230 9 ) ) PUNCT ejde-39 230 10 ) ) PUNCT ejde-39 230 11 , , PUNCT ejde-39 230 12 ( ( PUNCT ejde-39 230 13 4.1 4.1 NUM ejde-39 230 14 ) ) PUNCT ejde-39 230 15 where where SCONJ ejde-39 230 16 γ γ X ejde-39 230 17 = = PRON ejde-39 230 18 { { PUNCT ejde-39 230 19 h h NOUN ejde-39 230 20 ∈ ∈ PROPN ejde-39 230 21 c(b;w c(b;w PROPN ejde-39 230 22 ) ) PUNCT ejde-39 230 23 : : PUNCT ejde-39 230 24 h|b0 h|b0 X ejde-39 230 25 = = SYM ejde-39 230 26 idb0 idb0 PROPN ejde-39 230 27 } } PUNCT ejde-39 230 28 . . PUNCT ejde-39 231 1 by by ADP ejde-39 231 2 propositions proposition NOUN ejde-39 231 3 3.3 3.3 NUM ejde-39 231 4 and and CCONJ ejde-39 231 5 corollary corollary ADJ ejde-39 231 6 2.3 2.3 NUM ejde-39 231 7 the the DET ejde-39 231 8 functionalreρ functionalreρ NOUN ejde-39 231 9 satisfies satisfy VERB ejde-39 231 10 the the DET ejde-39 231 11 ( ( PUNCT ejde-39 231 12 ce ce PROPN ejde-39 231 13 ) ) PUNCT ejde-39 231 14 condition condition NOUN ejde-39 231 15 at at ADP ejde-39 231 16 the the DET ejde-39 231 17 level level NOUN ejde-39 231 18 c c NOUN ejde-39 231 19 = = SYM ejde-39 231 20 µkλ(e µkλ(e PROPN ejde-39 231 21 ) ) PUNCT ejde-39 231 22 > > X ejde-39 231 23 0 0 X ejde-39 231 24 . . PUNCT ejde-39 232 1 note note VERB ejde-39 232 2 that that SCONJ ejde-39 232 3 , , PUNCT ejde-39 232 4 for for ADP ejde-39 232 5 t t PROPN ejde-39 232 6 > > X ejde-39 232 7 rkλ rkλ PROPN ejde-39 232 8 , , PUNCT ejde-39 232 9 b0∩s+ b0∩s+ ADV ejde-39 232 10 rkλ rkλ NOUN ejde-39 232 11 = = SYM ejde-39 232 12 ∅ ∅ NOUN ejde-39 232 13 , , PUNCT ejde-39 232 14 d(b0 d(b0 NOUN ejde-39 232 15 , , PUNCT ejde-39 232 16 s s PART ejde-39 232 17 + + NUM ejde-39 232 18 rkλ rkλ NOUN ejde-39 232 19 ) ) PUNCT ejde-39 232 20 > > X ejde-39 232 21 0 0 NUM ejde-39 232 22 , , PUNCT ejde-39 232 23 and and CCONJ ejde-39 232 24 s+ s+ ADV ejde-39 232 25 rkλ rkλ NOUN ejde-39 232 26 is be AUX ejde-39 232 27 closed close VERB ejde-39 232 28 in in ADP ejde-39 232 29 w w PROPN ejde-39 232 30 . . PUNCT ejde-39 233 1 furthermore furthermore ADV ejde-39 233 2 , , PUNCT ejde-39 233 3 it it PRON ejde-39 233 4 can can AUX ejde-39 233 5 be be AUX ejde-39 233 6 shown show VERB ejde-39 233 7 in in ADP ejde-39 233 8 a a DET ejde-39 233 9 standard standard ADJ ejde-39 233 10 way way NOUN ejde-39 233 11 ( ( PUNCT ejde-39 233 12 see see VERB ejde-39 233 13 [ [ X ejde-39 233 14 25 25 NUM ejde-39 233 15 , , PUNCT ejde-39 233 16 p. p. NOUN ejde-39 233 17 156 156 NUM ejde-39 233 18 ] ] PUNCT ejde-39 233 19 ) ) PUNCT ejde-39 233 20 that that SCONJ ejde-39 233 21 { { PUNCT ejde-39 233 22 b0 b0 NOUN ejde-39 233 23 , , PUNCT ejde-39 233 24 b b NOUN ejde-39 233 25 } } PUNCT ejde-39 233 26 links link NOUN ejde-39 233 27 s+ s+ ADV ejde-39 233 28 rkλ rkλ NOUN ejde-39 233 29 in in ADP ejde-39 233 30 w w PROPN ejde-39 233 31 . . PUNCT ejde-39 234 1 hence hence ADV ejde-39 234 2 , , PUNCT ejde-39 234 3 by by ADP ejde-39 234 4 the the DET ejde-39 234 5 benci benci PROPN ejde-39 234 6 - - PUNCT ejde-39 234 7 rabinowitz rabinowitz PROPN ejde-39 234 8 linking linking NOUN ejde-39 234 9 theorem theorem ADJ ejde-39 234 10 [ [ X ejde-39 234 11 5 5 NUM ejde-39 234 12 ] ] PUNCT ejde-39 234 13 for for ADP ejde-39 234 14 functionals functional NOUN ejde-39 234 15 satisfying satisfying ADJ ejde-39 234 16 ( ( PUNCT ejde-39 234 17 ce ce PROPN ejde-39 234 18 ) ) PUNCT ejde-39 234 19 condition condition NOUN ejde-39 234 20 ( ( PUNCT ejde-39 234 21 see see VERB ejde-39 234 22 [ [ X ejde-39 234 23 25 25 NUM ejde-39 234 24 , , PUNCT ejde-39 234 25 theorem theorem VERB ejde-39 234 26 5.39 5.39 NUM ejde-39 234 27 ] ] PUNCT ejde-39 234 28 and and CCONJ ejde-39 234 29 appendix appendix VERB ejde-39 234 30 below below ADV ejde-39 234 31 ) ) PUNCT ejde-39 234 32 , , PUNCT ejde-39 234 33 there there PRON ejde-39 234 34 exists exist VERB ejde-39 234 35 a a DET ejde-39 234 36 nonzero nonzero ADJ ejde-39 234 37 critical critical ADJ ejde-39 234 38 point point NOUN ejde-39 234 39 uµkλ(e uµkλ(e ADJ ejde-39 234 40 ) ) PUNCT ejde-39 234 41 ∈ ∈ PROPN ejde-39 234 42 w w PROPN ejde-39 234 43 \ \ PROPN ejde-39 234 44 { { PUNCT ejde-39 234 45 0 0 NUM ejde-39 234 46 } } PUNCT ejde-39 234 47 of of ADP ejde-39 234 48 the the DET ejde-39 234 49 functional functional ADJ ejde-39 234 50 reρ reρ NOUN ejde-39 234 51 such such ADJ ejde-39 234 52 that that DET ejde-39 234 53 reρ reρ NOUN ejde-39 234 54 ( ( PUNCT ejde-39 234 55 uµkλ(e uµkλ(e NOUN ejde-39 234 56 ) ) PUNCT ejde-39 234 57 ) ) PUNCT ejde-39 234 58 = = SYM ejde-39 235 1 µkλ(e µkλ(e VERB ejde-39 235 2 ) ) PUNCT ejde-39 235 3 ≥ ≥ NOUN ejde-39 235 4 a a DET ejde-39 235 5 > > X ejde-39 235 6 0 0 X ejde-39 235 7 . . PUNCT ejde-39 236 1 consequently consequently ADV ejde-39 236 2 , , PUNCT ejde-39 236 3 corollary corollary ADJ ejde-39 236 4 2.2 2.2 NUM ejde-39 236 5 and and CCONJ ejde-39 236 6 ( ( PUNCT ejde-39 236 7 1.3 1.3 NUM ejde-39 236 8 ) ) PUNCT ejde-39 236 9 yields yield NOUN ejde-39 236 10 that that PRON ejde-39 236 11 uµkλ(e uµkλ(e NOUN ejde-39 236 12 ) ) PUNCT ejde-39 236 13 is be AUX ejde-39 236 14 a a DET ejde-39 236 15 weak weak ADJ ejde-39 236 16 solution solution NOUN ejde-39 236 17 of of ADP ejde-39 236 18 ( ( PUNCT ejde-39 236 19 1.1 1.1 NUM ejde-39 236 20 ) ) PUNCT ejde-39 236 21 with with ADP ejde-39 236 22 µ µ NOUN ejde-39 236 23 = = SYM ejde-39 236 24 µkλ(e µkλ(e VERB ejde-39 236 25 ) ) PUNCT ejde-39 236 26 and and CCONJ ejde-39 236 27 energy energy NOUN ejde-39 236 28 value value NOUN ejde-39 236 29 e. e. PROPN ejde-39 236 30 standard standard PROPN ejde-39 236 31 bootstrap bootstrap NOUN ejde-39 236 32 arguments argument NOUN ejde-39 236 33 and and CCONJ ejde-39 236 34 sobolev sobolev NOUN ejde-39 236 35 ’s ’s PART ejde-39 236 36 embedding embed VERB ejde-39 236 37 theorem theorem NOUN ejde-39 236 38 ( ( PUNCT ejde-39 236 39 see see VERB ejde-39 236 40 , , PUNCT ejde-39 236 41 e.g. e.g. ADV ejde-39 236 42 , , PUNCT ejde-39 236 43 [ [ X ejde-39 236 44 28 28 NUM ejde-39 236 45 ] ] PUNCT ejde-39 236 46 ) ) PUNCT ejde-39 236 47 entail entail NOUN ejde-39 236 48 that that PRON ejde-39 236 49 uµkλ(e uµkλ(e NOUN ejde-39 236 50 ) ) PUNCT ejde-39 236 51 ∈ ∈ NOUN ejde-39 236 52 l∞(ω l∞(ω NOUN ejde-39 236 53 ) ) PUNCT ejde-39 236 54 . . PUNCT ejde-39 237 1 therefore therefore ADV ejde-39 237 2 , , PUNCT ejde-39 237 3 by by ADP ejde-39 237 4 the the DET ejde-39 237 5 lp lp ADJ ejde-39 237 6 - - PUNCT ejde-39 237 7 regularity regularity NOUN ejde-39 237 8 results result NOUN ejde-39 237 9 in in ADP ejde-39 237 10 [ [ X ejde-39 237 11 16 16 NUM ejde-39 237 12 ] ] PUNCT ejde-39 237 13 , , PUNCT ejde-39 237 14 uµkλ(e uµkλ(e ADJ ejde-39 237 15 ) ) PUNCT ejde-39 237 16 ∈ ∈ PROPN ejde-39 237 17 w w PROPN ejde-39 237 18 2,p(ω 2,p(ω PROPN ejde-39 237 19 ) ) PUNCT ejde-39 237 20 for for ADP ejde-39 237 21 any any DET ejde-39 237 22 1 1 NUM ejde-39 237 23 < < X ejde-39 237 24 p p X ejde-39 237 25 < < X ejde-39 237 26 ∞ ∞ PROPN ejde-39 237 27 and and CCONJ ejde-39 237 28 thus thus ADV ejde-39 237 29 , , PUNCT ejde-39 237 30 by by ADP ejde-39 237 31 sobolev sobolev NOUN ejde-39 237 32 ’s ’s PART ejde-39 237 33 embedding embed VERB ejde-39 237 34 theorem theorem NOUN ejde-39 237 35 , , PUNCT ejde-39 237 36 uµkλ(e uµkλ(e ADJ ejde-39 237 37 ) ) PUNCT ejde-39 237 38 ∈ ∈ NOUN ejde-39 237 39 c1,α(ω c1,α(ω NOUN ejde-39 237 40 ) ) PUNCT ejde-39 237 41 for for ADP ejde-39 237 42 any any DET ejde-39 237 43 α α NOUN ejde-39 237 44 ∈ ∈ PROPN ejde-39 237 45 ( ( PUNCT ejde-39 237 46 0 0 NUM ejde-39 237 47 , , PUNCT ejde-39 237 48 1 1 NUM ejde-39 237 49 ) ) PUNCT ejde-39 237 50 . . PUNCT ejde-39 238 1 this this PRON ejde-39 238 2 completes complete VERB ejde-39 238 3 the the DET ejde-39 238 4 proof proof NOUN ejde-39 238 5 of of ADP ejde-39 238 6 the the DET ejde-39 238 7 first first ADJ ejde-39 238 8 part part NOUN ejde-39 238 9 of of ADP ejde-39 238 10 the the DET ejde-39 238 11 theorem theorem NOUN ejde-39 238 12 . . PUNCT ejde-39 239 1 ( ( PUNCT ejde-39 239 2 i i NOUN ejde-39 239 3 ) ) PUNCT ejde-39 239 4 take take VERB ejde-39 239 5 e1 e1 PROPN ejde-39 239 6 > > X ejde-39 239 7 e0 e0 PROPN ejde-39 239 8 > > X ejde-39 239 9 0 0 X ejde-39 239 10 . . PUNCT ejde-39 240 1 it it PRON ejde-39 240 2 is be AUX ejde-39 240 3 not not PART ejde-39 240 4 hard hard ADJ ejde-39 240 5 to to PART ejde-39 240 6 see see VERB ejde-39 240 7 that that SCONJ ejde-39 240 8 the the DET ejde-39 240 9 sets set NOUN ejde-39 240 10 { { PUNCT ejde-39 240 11 b0 b0 NOUN ejde-39 240 12 , , PUNCT ejde-39 240 13 b b NOUN ejde-39 240 14 } } PUNCT ejde-39 240 15 and and CCONJ ejde-39 240 16 the the DET ejde-39 240 17 path path NOUN ejde-39 240 18 sets set VERB ejde-39 240 19 γ γ NOUN ejde-39 240 20 in in ADP ejde-39 240 21 ( ( PUNCT ejde-39 240 22 4.1 4.1 NUM ejde-39 240 23 ) ) PUNCT ejde-39 240 24 can can AUX ejde-39 240 25 be be AUX ejde-39 240 26 taken take VERB ejde-39 240 27 the the DET ejde-39 240 28 same same ADJ ejde-39 240 29 for for ADP ejde-39 240 30 e1 e1 NOUN ejde-39 240 31 , , PUNCT ejde-39 240 32 e0 e0 PROPN ejde-39 240 33 if if SCONJ ejde-39 240 34 |e1 |e1 NOUN ejde-39 241 1 − − PROPN ejde-39 241 2 e0| e0| VERB ejde-39 241 3 is be AUX ejde-39 241 4 sufficiently sufficiently ADV ejde-39 241 5 small small ADJ ejde-39 241 6 . . PUNCT ejde-39 242 1 note note VERB ejde-39 242 2 that that SCONJ ejde-39 242 3 re1 re1 PROPN ejde-39 242 4 ρ ρ PROPN ejde-39 242 5 ( ( PUNCT ejde-39 242 6 u u NOUN ejde-39 242 7 ) ) PUNCT ejde-39 242 8 = = SYM ejde-39 242 9 re0 re0 PROPN ejde-39 242 10 ρ ρ PROPN ejde-39 242 11 ( ( PUNCT ejde-39 242 12 u)− u)− PROPN ejde-39 242 13 φρ(‖u‖1 φρ(‖u‖1 NOUN ejde-39 242 14 ) ) PUNCT ejde-39 242 15 e1 e1 PROPN ejde-39 242 16 − − PROPN ejde-39 242 17 e0∫ e0∫ PROPN ejde-39 242 18 g(x g(x NOUN ejde-39 242 19 , , PUNCT ejde-39 242 20 u u NOUN ejde-39 242 21 ) ) PUNCT ejde-39 242 22 dx dx PROPN ejde-39 242 23 , , PUNCT ejde-39 242 24 ∀u ∀u NOUN ejde-39 242 25 ∈w ∈w VERB ejde-39 242 26 \ \ NOUN ejde-39 242 27 0 0 NUM ejde-39 242 28 , , PUNCT ejde-39 242 29 and and CCONJ ejde-39 242 30 thus thus ADV ejde-39 242 31 max max PROPN ejde-39 242 32 u∈b u∈b PROPN ejde-39 242 33 re1 re1 PROPN ejde-39 242 34 ρ ρ PROPN ejde-39 242 35 ( ( PUNCT ejde-39 242 36 h(u h(u PROPN ejde-39 242 37 ) ) PUNCT ejde-39 242 38 ) ) PUNCT ejde-39 243 1 = = SYM ejde-39 243 2 max max PROPN ejde-39 243 3 u∈b u∈b NOUN ejde-39 243 4 ( ( PUNCT ejde-39 243 5 re0 re0 PROPN ejde-39 243 6 ρ ρ PROPN ejde-39 243 7 ( ( PUNCT ejde-39 243 8 h(u))− h(u))− NOUN ejde-39 243 9 φρ(‖h(u)‖1 φρ(‖h(u)‖1 NOUN ejde-39 243 10 ) ) PUNCT ejde-39 243 11 e1 e1 PROPN ejde-39 243 12 − − PROPN ejde-39 243 13 e0∫ e0∫ PROPN ejde-39 243 14 g(x g(x NOUN ejde-39 243 15 , , PUNCT ejde-39 243 16 h(u h(u PROPN ejde-39 243 17 ) ) PUNCT ejde-39 243 18 ) ) PUNCT ejde-39 243 19 dx dx PROPN ejde-39 243 20 ) ) PUNCT ejde-39 243 21 ≤ ≤ NUM ejde-39 244 1 max max PROPN ejde-39 244 2 u∈b u∈b PROPN ejde-39 244 3 re0 re0 PROPN ejde-39 244 4 ρ ρ PROPN ejde-39 244 5 ( ( PUNCT ejde-39 244 6 h(u h(u PROPN ejde-39 244 7 ) ) PUNCT ejde-39 244 8 ) ) PUNCT ejde-39 244 9 , , PUNCT ejde-39 244 10 ∀h ∀h PROPN ejde-39 244 11 ∈ ∈ PROPN ejde-39 244 12 γ γ X ejde-39 244 13 , , PUNCT ejde-39 244 14 and and CCONJ ejde-39 244 15 therefore therefore ADV ejde-39 244 16 , , PUNCT ejde-39 244 17 for for ADP ejde-39 244 18 sufficiently sufficiently ADV ejde-39 244 19 small small ADJ ejde-39 244 20 |e1 |e1 NOUN ejde-39 244 21 − − PROPN ejde-39 244 22 e0| e0| NOUN ejde-39 244 23 , , PUNCT ejde-39 244 24 we we PRON ejde-39 244 25 have have VERB ejde-39 244 26 µkλ(e1 µkλ(e1 NOUN ejde-39 244 27 ) ) PUNCT ejde-39 245 1 = = SYM ejde-39 245 2 inf inf PROPN ejde-39 245 3 h∈γ h∈γ NOUN ejde-39 245 4 max max PROPN ejde-39 245 5 u∈b u∈b PROPN ejde-39 245 6 re1 re1 PROPN ejde-39 245 7 ρ ρ PROPN ejde-39 245 8 ( ( PUNCT ejde-39 245 9 h(u h(u PROPN ejde-39 245 10 ) ) PUNCT ejde-39 245 11 ) ) PUNCT ejde-39 245 12 ≤ ≤ NUM ejde-39 246 1 inf inf PROPN ejde-39 246 2 h∈γ h∈γ NOUN ejde-39 246 3 max max PROPN ejde-39 246 4 u∈b u∈b PROPN ejde-39 246 5 re0 re0 PROPN ejde-39 246 6 ρ ρ PROPN ejde-39 246 7 ( ( PUNCT ejde-39 246 8 h(u h(u PROPN ejde-39 246 9 ) ) PUNCT ejde-39 246 10 ) ) PUNCT ejde-39 247 1 = = PUNCT ejde-39 247 2 µkλ(e0 µkλ(e0 NOUN ejde-39 247 3 ) ) PUNCT ejde-39 247 4 . . PUNCT ejde-39 248 1 ( ( PUNCT ejde-39 248 2 ii ii NOUN ejde-39 248 3 ) ) PUNCT ejde-39 248 4 let let VERB ejde-39 248 5 e e NOUN ejde-39 248 6 = = SYM ejde-39 248 7 0 0 X ejde-39 248 8 . . PUNCT ejde-39 248 9 consider consider VERB ejde-39 249 1 r0(u r0(u NUM ejde-39 249 2 ) ) PUNCT ejde-39 249 3 ≡ ≡ PROPN ejde-39 249 4 re(u)|e=0 re(u)|e=0 NOUN ejde-39 249 5 . . PUNCT ejde-39 250 1 using use VERB ejde-39 250 2 ( ( PUNCT ejde-39 250 3 3.1 3.1 NUM ejde-39 250 4 ) ) PUNCT ejde-39 250 5 and and CCONJ ejde-39 250 6 1 1 NUM ejde-39 250 7 < < X ejde-39 250 8 q q X ejde-39 250 9 < < X ejde-39 250 10 2 2 NUM ejde-39 250 11 it it PRON ejde-39 250 12 is be AUX ejde-39 250 13 not not PART ejde-39 250 14 hard hard ADJ ejde-39 250 15 to to PART ejde-39 250 16 show show VERB ejde-39 250 17 that that SCONJ ejde-39 250 18 r0(u)→ r0(u)→ NOUN ejde-39 250 19 0 0 NUM ejde-39 250 20 as as ADP ejde-39 250 21 ‖u‖1 ‖u‖1 NOUN ejde-39 250 22 → → SYM ejde-39 250 23 0 0 NUM ejde-39 250 24 . . PUNCT ejde-39 251 1 consequently consequently ADV ejde-39 251 2 , , PUNCT ejde-39 251 3 by by ADP ejde-39 251 4 the the DET ejde-39 251 5 continuation continuation NOUN ejde-39 251 6 we we PRON ejde-39 251 7 can can AUX ejde-39 251 8 set set VERB ejde-39 251 9 that that DET ejde-39 251 10 r0(0 r0(0 NOUN ejde-39 251 11 ) ) PUNCT ejde-39 251 12 = = SYM ejde-39 252 1 0 0 X ejde-39 252 2 . . PUNCT ejde-39 253 1 ejde-2023/23 ejde-2023/23 NOUN ejde-39 253 2 prescribed prescribe VERB ejde-39 253 3 energy energy NOUN ejde-39 253 4 saddle saddle NOUN ejde-39 253 5 - - PUNCT ejde-39 253 6 point point NOUN ejde-39 253 7 solutions solution NOUN ejde-39 253 8 9 9 NUM ejde-39 253 9 assume assume VERB ejde-39 253 10 that that SCONJ ejde-39 253 11 λ λ PROPN ejde-39 253 12 < < X ejde-39 253 13 λ1 λ1 PROPN ejde-39 253 14 . . PUNCT ejde-39 254 1 then then ADV ejde-39 254 2 w− w− NOUN ejde-39 254 3 = = SYM ejde-39 254 4 ∅ ∅ NOUN ejde-39 254 5 and and CCONJ ejde-39 254 6 w+ w+ NOUN ejde-39 254 7 ≡w ≡w NOUN ejde-39 254 8 . . PUNCT ejde-39 255 1 by by ADP ejde-39 255 2 proposition proposition NOUN ejde-39 255 3 3.2 3.2 NUM ejde-39 255 4 , , PUNCT ejde-39 255 5 one one PRON ejde-39 255 6 can can AUX ejde-39 255 7 find find VERB ejde-39 255 8 u1 u1 NOUN ejde-39 255 9 ∈ ∈ PROPN ejde-39 255 10 w w ADP ejde-39 255 11 such such ADJ ejde-39 255 12 that that DET ejde-39 255 13 re(u1 re(u1 NOUN ejde-39 255 14 ) ) PUNCT ejde-39 255 15 < < X ejde-39 255 16 0 0 X ejde-39 255 17 . . PUNCT ejde-39 256 1 let let VERB ejde-39 256 2 e e X ejde-39 256 3 ∈ ∈ PROPN ejde-39 257 1 [ [ X ejde-39 257 2 0 0 NUM ejde-39 257 3 , , PUNCT ejde-39 257 4 e0 e0 PROPN ejde-39 257 5 λ λ PROPN ejde-39 257 6 ) ) PUNCT ejde-39 257 7 and and CCONJ ejde-39 257 8 0 0 NUM ejde-39 257 9 < < X ejde-39 257 10 ρ ρ X ejde-39 257 11 < < X ejde-39 257 12 min{rλ0 min{rλ0 X ejde-39 257 13 , , PUNCT ejde-39 257 14 ρ(e ρ(e PROPN ejde-39 257 15 ) ) PUNCT ejde-39 257 16 } } PUNCT ejde-39 257 17 . . PUNCT ejde-39 258 1 observe observe VERB ejde-39 258 2 that that SCONJ ejde-39 258 3 ( ( PUNCT ejde-39 258 4 4.1 4.1 NUM ejde-39 258 5 ) ) PUNCT ejde-39 258 6 can can AUX ejde-39 258 7 be be AUX ejde-39 258 8 rewritten rewrite VERB ejde-39 258 9 as as SCONJ ejde-39 258 10 follows follow VERB ejde-39 258 11 µ0 µ0 NOUN ejde-39 258 12 λ(e λ(e ADJ ejde-39 258 13 ) ) PUNCT ejde-39 258 14 : : PUNCT ejde-39 258 15 = = PROPN ejde-39 258 16 inf inf PROPN ejde-39 258 17 γ∈γ γ∈γ ADJ ejde-39 258 18 max max PROPN ejde-39 258 19 t∈[0,1 t∈[0,1 PROPN ejde-39 258 20 ] ] X ejde-39 258 21 reρ reρ NOUN ejde-39 258 22 ( ( PUNCT ejde-39 258 23 γ(t γ(t NOUN ejde-39 258 24 ) ) PUNCT ejde-39 258 25 ) ) PUNCT ejde-39 258 26 , , PUNCT ejde-39 258 27 ( ( PUNCT ejde-39 258 28 4.2 4.2 NUM ejde-39 258 29 ) ) PUNCT ejde-39 258 30 where where SCONJ ejde-39 258 31 γ γ X ejde-39 258 32 = = PRON ejde-39 258 33 { { PUNCT ejde-39 258 34 γ γ X ejde-39 258 35 ∈ ∈ PROPN ejde-39 258 36 c([0 c([0 NOUN ejde-39 258 37 , , PUNCT ejde-39 258 38 1];w 1];w NUM ejde-39 258 39 ) ) PUNCT ejde-39 258 40 : : PUNCT ejde-39 258 41 γ(0 γ(0 PROPN ejde-39 258 42 ) ) PUNCT ejde-39 258 43 = = SYM ejde-39 258 44 0 0 NUM ejde-39 258 45 , , PUNCT ejde-39 258 46 γ(1 γ(1 PROPN ejde-39 258 47 ) ) PUNCT ejde-39 258 48 = = SYM ejde-39 258 49 u1 u1 NOUN ejde-39 258 50 } } PUNCT ejde-39 258 51 , , PUNCT ejde-39 258 52 0 0 NUM ejde-39 258 53 < < X ejde-39 258 54 ρ ρ X ejde-39 258 55 < < X ejde-39 258 56 ρ(e ρ(e PROPN ejde-39 258 57 ) ) PUNCT ejde-39 258 58 . . PUNCT ejde-39 259 1 here here ADV ejde-39 259 2 we we PRON ejde-39 259 3 set set VERB ejde-39 259 4 r0 r0 NOUN ejde-39 259 5 ρ(u ρ(u PROPN ejde-39 259 6 ) ) PUNCT ejde-39 259 7 : : PUNCT ejde-39 260 1 = = PUNCT ejde-39 260 2 r0(u r0(u NUM ejde-39 260 3 ) ) PUNCT ejde-39 260 4 , , PUNCT ejde-39 260 5 ρ ρ PROPN ejde-39 260 6 > > X ejde-39 260 7 0 0 PROPN ejde-39 260 8 . . PUNCT ejde-39 260 9 note note VERB ejde-39 260 10 that that SCONJ ejde-39 260 11 by by ADP ejde-39 260 12 the the DET ejde-39 260 13 above above ADJ ejde-39 260 14 , , PUNCT ejde-39 260 15 for for ADP ejde-39 260 16 any any DET ejde-39 260 17 e e NOUN ejde-39 260 18 ∈ ∈ PROPN ejde-39 260 19 ( ( PUNCT ejde-39 260 20 0 0 NUM ejde-39 260 21 , , PUNCT ejde-39 260 22 e0 e0 PROPN ejde-39 260 23 λ λ PROPN ejde-39 260 24 ) ) PUNCT ejde-39 260 25 and and CCONJ ejde-39 260 26 0 0 NUM ejde-39 260 27 < < X ejde-39 260 28 ρ ρ X ejde-39 260 29 < < X ejde-39 260 30 min{rλ0 min{rλ0 X ejde-39 260 31 , , PUNCT ejde-39 260 32 ρ(e ρ(e PROPN ejde-39 260 33 ) ) PUNCT ejde-39 260 34 } } PUNCT ejde-39 260 35 , , PUNCT ejde-39 260 36 µ0 µ0 NOUN ejde-39 260 37 λ(e λ(e VERB ejde-39 260 38 ) ) PUNCT ejde-39 260 39 > > X ejde-39 260 40 0 0 PUNCT ejde-39 260 41 and and CCONJ ejde-39 260 42 there there PRON ejde-39 260 43 exists exist VERB ejde-39 260 44 a a DET ejde-39 260 45 critical critical ADJ ejde-39 260 46 point point NOUN ejde-39 260 47 uµ0 uµ0 NOUN ejde-39 260 48 λ(e λ(e NOUN ejde-39 260 49 ) ) PUNCT ejde-39 260 50 ∈ ∈ PROPN ejde-39 260 51 w w NOUN ejde-39 260 52 \ \ PROPN ejde-39 260 53 0 0 NUM ejde-39 260 54 of of ADP ejde-39 260 55 re(u re(u NOUN ejde-39 260 56 ) ) PUNCT ejde-39 260 57 such such ADJ ejde-39 260 58 that that DET ejde-39 260 59 deµ0 deµ0 PROPN ejde-39 260 60 λ(e)(uµ0 λ(e)(uµ0 PUNCT ejde-39 260 61 λ(e λ(e VERB ejde-39 260 62 ) ) PUNCT ejde-39 260 63 ) ) PUNCT ejde-39 261 1 = = SYM ejde-39 261 2 0 0 NUM ejde-39 261 3 and and CCONJ ejde-39 261 4 eµ0 eµ0 NUM ejde-39 261 5 λ(e)(uµ0 λ(e)(uµ0 PUNCT ejde-39 261 6 λ(e λ(e ADJ ejde-39 261 7 ) ) PUNCT ejde-39 261 8 ) ) PUNCT ejde-39 262 1 = = SYM ejde-39 262 2 e. e. PROPN ejde-39 262 3 as as ADP ejde-39 262 4 in in ADP ejde-39 262 5 the the DET ejde-39 262 6 proof proof NOUN ejde-39 262 7 of of ADP ejde-39 262 8 ( ( PUNCT ejde-39 262 9 i i PROPN ejde-39 262 10 ) ) PUNCT ejde-39 262 11 , , PUNCT ejde-39 262 12 from from ADP ejde-39 262 13 ( ( PUNCT ejde-39 262 14 4.2 4.2 NUM ejde-39 262 15 ) ) PUNCT ejde-39 262 16 it it PRON ejde-39 262 17 follows follow VERB ejde-39 262 18 that that SCONJ ejde-39 262 19 µ0 µ0 PROPN ejde-39 262 20 λ(e λ(e VERB ejde-39 262 21 ) ) PUNCT ejde-39 262 22 is be AUX ejde-39 262 23 a a DET ejde-39 262 24 non non ADJ ejde-39 262 25 - - ADJ ejde-39 262 26 increasing increasing ADJ ejde-39 262 27 function function NOUN ejde-39 262 28 on on ADP ejde-39 262 29 e e PROPN ejde-39 262 30 ∈ ∈ PROPN ejde-39 263 1 [ [ X ejde-39 263 2 0 0 NUM ejde-39 263 3 , , PUNCT ejde-39 263 4 e0 e0 PROPN ejde-39 263 5 λ λ PROPN ejde-39 263 6 ) ) PUNCT ejde-39 263 7 . . PUNCT ejde-39 264 1 moreover moreover ADV ejde-39 264 2 , , PUNCT ejde-39 264 3 µ0 µ0 NOUN ejde-39 264 4 λ(e λ(e ADJ ejde-39 264 5 ) ) PUNCT ejde-39 264 6 ≤ ≤ NOUN ejde-39 264 7 µ0 µ0 NOUN ejde-39 264 8 λ(0 λ(0 PROPN ejde-39 264 9 ) ) PUNCT ejde-39 264 10 < < X ejde-39 265 1 + + PROPN ejde-39 265 2 ∞ ∞ PROPN ejde-39 265 3 , , PUNCT ejde-39 265 4 for for ADP ejde-39 265 5 any any DET ejde-39 265 6 e e PROPN ejde-39 265 7 ∈ ∈ PROPN ejde-39 265 8 ( ( PUNCT ejde-39 265 9 0 0 NUM ejde-39 265 10 , , PUNCT ejde-39 265 11 e0 e0 PROPN ejde-39 265 12 λ λ PROPN ejde-39 265 13 ) ) PUNCT ejde-39 265 14 . . PUNCT ejde-39 266 1 hence hence ADV ejde-39 266 2 there there PRON ejde-39 266 3 exists exist VERB ejde-39 266 4 lime→0 lime→0 PROPN ejde-39 266 5 µ µ X ejde-39 266 6 0 0 NUM ejde-39 266 7 λ(e λ(e ADJ ejde-39 266 8 ) ) PUNCT ejde-39 266 9 = = SYM ejde-39 266 10 µ̄λ(0 µ̄λ(0 NOUN ejde-39 266 11 ) ) PUNCT ejde-39 266 12 ≤ ≤ NOUN ejde-39 266 13 µ0 µ0 NOUN ejde-39 266 14 λ(0 λ(0 PROPN ejde-39 266 15 ) ) PUNCT ejde-39 266 16 . . PUNCT ejde-39 267 1 furthermore furthermore ADV ejde-39 267 2 , , PUNCT ejde-39 267 3 µ̄λ(0 µ̄λ(0 NOUN ejde-39 267 4 ) ) PUNCT ejde-39 267 5 > > X ejde-39 267 6 0 0 PUNCT ejde-39 267 7 since since SCONJ ejde-39 267 8 µ0 µ0 NOUN ejde-39 267 9 λ(e λ(e VERB ejde-39 267 10 ) ) PUNCT ejde-39 267 11 > > X ejde-39 267 12 0 0 NUM ejde-39 267 13 , , PUNCT ejde-39 267 14 e e PROPN ejde-39 267 15 ∈ ∈ PROPN ejde-39 267 16 ( ( PUNCT ejde-39 267 17 0 0 NUM ejde-39 267 18 , , PUNCT ejde-39 267 19 ekλ ekλ ADJ ejde-39 267 20 ) ) PUNCT ejde-39 267 21 and and CCONJ ejde-39 267 22 µ0 µ0 NOUN ejde-39 267 23 λ(e λ(e VERB ejde-39 267 24 ) ) PUNCT ejde-39 267 25 is be AUX ejde-39 267 26 a a DET ejde-39 267 27 non non ADJ ejde-39 267 28 - - ADJ ejde-39 267 29 increasing increasing ADJ ejde-39 267 30 function function NOUN ejde-39 267 31 . . PUNCT ejde-39 268 1 since since SCONJ ejde-39 268 2 dre(uµ0 dre(uµ0 PROPN ejde-39 268 3 λ(e λ(e NUM ejde-39 268 4 ) ) PUNCT ejde-39 268 5 ) ) PUNCT ejde-39 268 6 = = SYM ejde-39 268 7 0 0 PUNCT ejde-39 268 8 and and CCONJ ejde-39 268 9 µ0 µ0 NOUN ejde-39 268 10 λ(e λ(e ADJ ejde-39 268 11 ) ) PUNCT ejde-39 268 12 ≡ ≡ PROPN ejde-39 268 13 re(uµ0 re(uµ0 PUNCT ejde-39 268 14 λ(e λ(e VERB ejde-39 268 15 ) ) PUNCT ejde-39 268 16 ) ) PUNCT ejde-39 268 17 → → SYM ejde-39 268 18 µ̄λ(0 µ̄λ(0 NOUN ejde-39 268 19 ) ) PUNCT ejde-39 268 20 > > X ejde-39 268 21 0 0 NUM ejde-39 268 22 , , PUNCT ejde-39 268 23 any any DET ejde-39 268 24 countable countable ADJ ejde-39 268 25 subset subset NOUN ejde-39 268 26 of of ADP ejde-39 268 27 ( ( PUNCT ejde-39 268 28 uµ0 uµ0 PROPN ejde-39 268 29 λ(e))e∈(0,e0 λ(e))e∈(0,e0 NOUN ejde-39 268 30 λ λ PROPN ejde-39 268 31 ) ) PUNCT ejde-39 268 32 is be AUX ejde-39 268 33 a a DET ejde-39 268 34 ( ( PUNCT ejde-39 268 35 ce ce NOUN ejde-39 268 36 ) ) PUNCT ejde-39 268 37 sequence sequence NOUN ejde-39 268 38 . . PUNCT ejde-39 269 1 hence hence ADV ejde-39 269 2 proposition proposition VERB ejde-39 269 3 3.3 3.3 NUM ejde-39 269 4 implies imply VERB ejde-39 269 5 that that SCONJ ejde-39 269 6 there there PRON ejde-39 269 7 exists exist VERB ejde-39 269 8 a a DET ejde-39 269 9 sequences sequence NOUN ejde-39 269 10 uµλ(em uµλ(em NUM ejde-39 269 11 ) ) PUNCT ejde-39 269 12 , , PUNCT ejde-39 269 13 m m VERB ejde-39 269 14 = = NOUN ejde-39 269 15 1 1 NUM ejde-39 269 16 , , PUNCT ejde-39 269 17 2 2 NUM ejde-39 269 18 , , PUNCT ejde-39 269 19 . . PUNCT ejde-39 269 20 . . PUNCT ejde-39 270 1 . . PUNCT ejde-39 271 1 , , PUNCT ejde-39 271 2 such such ADJ ejde-39 271 3 that that SCONJ ejde-39 271 4 limm→+∞em limm→+∞em ADJ ejde-39 271 5 = = SYM ejde-39 271 6 0 0 NUM ejde-39 271 7 and and CCONJ ejde-39 271 8 uµλ(em uµλ(em NOUN ejde-39 271 9 ) ) PUNCT ejde-39 271 10 convergences convergence NOUN ejde-39 271 11 in in ADP ejde-39 271 12 w w NOUN ejde-39 271 13 to to ADP ejde-39 271 14 some some DET ejde-39 271 15 point point NOUN ejde-39 271 16 uµ̄λ(0 uµ̄λ(0 PROPN ejde-39 271 17 ) ) PUNCT ejde-39 271 18 ∈ ∈ PROPN ejde-39 271 19 w w NOUN ejde-39 271 20 as as ADP ejde-39 271 21 m m PROPN ejde-39 271 22 → → SYM ejde-39 271 23 + + NUM ejde-39 271 24 ∞. ∞. PROPN ejde-39 271 25 note note VERB ejde-39 271 26 that that SCONJ ejde-39 271 27 uµ̄λ(0 uµ̄λ(0 PROPN ejde-39 271 28 ) ) PUNCT ejde-39 271 29 6= 6= ADP ejde-39 271 30 0 0 NUM ejde-39 272 1 ( ( PUNCT ejde-39 272 2 see see VERB ejde-39 272 3 remark remark NOUN ejde-39 272 4 3.4 3.4 NUM ejde-39 272 5 ) ) PUNCT ejde-39 272 6 , , PUNCT ejde-39 272 7 and and CCONJ ejde-39 272 8 therefore therefore ADV ejde-39 272 9 uµ̄λ(0 uµ̄λ(0 PROPN ejde-39 272 10 ) ) PUNCT ejde-39 272 11 is be AUX ejde-39 272 12 a a DET ejde-39 272 13 weak weak ADJ ejde-39 272 14 solution solution NOUN ejde-39 272 15 of of ADP ejde-39 272 16 ( ( PUNCT ejde-39 272 17 1.1 1.1 NUM ejde-39 272 18 ) ) PUNCT ejde-39 272 19 with with ADP ejde-39 272 20 µ µ NOUN ejde-39 272 21 = = SYM ejde-39 272 22 µ̄λ(0 µ̄λ(0 NOUN ejde-39 272 23 ) ) PUNCT ejde-39 272 24 . . PUNCT ejde-39 273 1 moreover moreover ADV ejde-39 273 2 , , PUNCT ejde-39 273 3 0 0 X ejde-39 273 4 = = SYM ejde-39 273 5 lim lim PROPN ejde-39 273 6 m→+∞ m→+∞ PROPN ejde-39 273 7 em em PRON ejde-39 273 8 = = SYM ejde-39 273 9 lim lim PROPN ejde-39 273 10 m→+∞ m→+∞ PROPN ejde-39 273 11 eλ,µ(uµλ(em eλ,µ(uµλ(em PROPN ejde-39 273 12 ) ) PUNCT ejde-39 273 13 ) ) PUNCT ejde-39 274 1 = = SYM ejde-39 274 2 eλ,µ(uµ̄λ(0 eλ,µ(uµ̄λ(0 PROPN ejde-39 274 3 ) ) PUNCT ejde-39 274 4 ) ) PUNCT ejde-39 274 5 . . PUNCT ejde-39 275 1 thus thus ADV ejde-39 275 2 uµ̄λ(0 uµ̄λ(0 PROPN ejde-39 275 3 ) ) PUNCT ejde-39 275 4 is be AUX ejde-39 275 5 a a DET ejde-39 275 6 solution solution NOUN ejde-39 275 7 with with ADP ejde-39 275 8 zero zero NUM ejde-39 275 9 energy energy NOUN ejde-39 275 10 . . PUNCT ejde-39 276 1 as as ADP ejde-39 276 2 above above ADP ejde-39 276 3 it it PRON ejde-39 276 4 can can AUX ejde-39 276 5 be be AUX ejde-39 276 6 shown show VERB ejde-39 276 7 that that SCONJ ejde-39 276 8 uµ̄λ(0 uµ̄λ(0 NOUN ejde-39 276 9 ) ) PUNCT ejde-39 276 10 ∈ ∈ PROPN ejde-39 276 11 c1,α(ω c1,α(ω NOUN ejde-39 276 12 ) ) PUNCT ejde-39 276 13 , , PUNCT ejde-39 276 14 α α PROPN ejde-39 276 15 ∈ ∈ PROPN ejde-39 276 16 ( ( PUNCT ejde-39 276 17 0 0 NUM ejde-39 276 18 , , PUNCT ejde-39 276 19 1 1 NUM ejde-39 276 20 ) ) PUNCT ejde-39 276 21 . . PUNCT ejde-39 277 1 5 5 X ejde-39 277 2 . . X ejde-39 277 3 conclusions conclusion NOUN ejde-39 277 4 and and CCONJ ejde-39 277 5 discussion discussion NOUN ejde-39 277 6 in in ADP ejde-39 277 7 this this DET ejde-39 277 8 paper paper NOUN ejde-39 277 9 , , PUNCT ejde-39 277 10 we we PRON ejde-39 277 11 develop develop VERB ejde-39 277 12 the the DET ejde-39 277 13 mountain mountain NOUN ejde-39 277 14 pass pass NOUN ejde-39 277 15 methods method NOUN ejde-39 277 16 applicable applicable ADJ ejde-39 277 17 to to ADP ejde-39 277 18 a a DET ejde-39 277 19 new new ADJ ejde-39 277 20 class class NOUN ejde-39 277 21 of of ADP ejde-39 277 22 problems problem NOUN ejde-39 277 23 . . PUNCT ejde-39 278 1 in in ADP ejde-39 278 2 particular particular ADJ ejde-39 278 3 , , PUNCT ejde-39 278 4 an an DET ejde-39 278 5 approach approach NOUN ejde-39 278 6 to to ADP ejde-39 278 7 finding find VERB ejde-39 278 8 mountain mountain NOUN ejde-39 278 9 pass pass NOUN ejde-39 278 10 - - PUNCT ejde-39 278 11 type type NOUN ejde-39 278 12 solutions solution NOUN ejde-39 278 13 with with ADP ejde-39 278 14 prescribed prescribed ADJ ejde-39 278 15 energy energy NOUN ejde-39 278 16 for for ADP ejde-39 278 17 indefinite indefinite ADJ ejde-39 278 18 elliptic elliptic ADJ ejde-39 278 19 problems problem NOUN ejde-39 278 20 with with ADP ejde-39 278 21 nonlinearities nonlinearitie NOUN ejde-39 278 22 which which PRON ejde-39 278 23 does do AUX ejde-39 278 24 not not PART ejde-39 278 25 satisfy satisfy VERB ejde-39 278 26 the the DET ejde-39 278 27 ambrosetti ambrosetti NOUN ejde-39 278 28 - - PUNCT ejde-39 278 29 rabinowitz rabinowitz NOUN ejde-39 278 30 growth growth NOUN ejde-39 278 31 conditions condition NOUN ejde-39 278 32 is be AUX ejde-39 278 33 introduced introduce VERB ejde-39 278 34 . . PUNCT ejde-39 279 1 furthermore furthermore ADV ejde-39 279 2 , , PUNCT ejde-39 279 3 the the DET ejde-39 279 4 method method NOUN ejde-39 279 5 of of ADP ejde-39 279 6 nonlinear nonlinear ADJ ejde-39 279 7 rayleigh rayleigh PROPN ejde-39 279 8 quotients quotient NOUN ejde-39 279 9 is be AUX ejde-39 279 10 used use VERB ejde-39 279 11 for for ADP ejde-39 279 12 the the DET ejde-39 279 13 first first ADJ ejde-39 279 14 time time NOUN ejde-39 279 15 to to PART ejde-39 279 16 solve solve VERB ejde-39 279 17 indefinite indefinite ADJ ejde-39 279 18 elliptic elliptic ADJ ejde-39 279 19 equations equation NOUN ejde-39 279 20 with with ADP ejde-39 279 21 general general ADJ ejde-39 279 22 forms form NOUN ejde-39 279 23 of of ADP ejde-39 279 24 non non NOUN ejde-39 279 25 - - NOUN ejde-39 279 26 linearities linearity NOUN ejde-39 279 27 . . PUNCT ejde-39 280 1 a a DET ejde-39 280 2 valuable valuable ADJ ejde-39 280 3 property property NOUN ejde-39 280 4 of of ADP ejde-39 280 5 the the DET ejde-39 280 6 nonlinear nonlinear ADJ ejde-39 280 7 rayleigh rayleigh PROPN ejde-39 280 8 quotients quotient NOUN ejde-39 280 9 method method NOUN ejde-39 280 10 is be AUX ejde-39 280 11 that that SCONJ ejde-39 280 12 it it PRON ejde-39 280 13 simplifies simplify VERB ejde-39 280 14 the the DET ejde-39 280 15 complexity complexity NOUN ejde-39 280 16 problem problem NOUN ejde-39 280 17 in in ADP ejde-39 280 18 a a DET ejde-39 280 19 sense sense NOUN ejde-39 280 20 by by ADP ejde-39 280 21 reducing reduce VERB ejde-39 280 22 degree degree NOUN ejde-39 280 23 of of ADP ejde-39 280 24 degeneracy degeneracy NOUN ejde-39 280 25 of of ADP ejde-39 280 26 the the DET ejde-39 280 27 system system NOUN ejde-39 280 28 ( ( PUNCT ejde-39 280 29 see see VERB ejde-39 280 30 [ [ X ejde-39 280 31 3 3 NUM ejde-39 280 32 , , PUNCT ejde-39 280 33 8 8 NUM ejde-39 280 34 ] ] NUM ejde-39 280 35 ) ) PUNCT ejde-39 280 36 . . PUNCT ejde-39 281 1 however however ADV ejde-39 281 2 , , PUNCT ejde-39 281 3 applicability applicability NOUN ejde-39 281 4 of of ADP ejde-39 281 5 general general ADJ ejde-39 281 6 theories theory NOUN ejde-39 281 7 like like ADP ejde-39 281 8 the the DET ejde-39 281 9 mountain mountain NOUN ejde-39 281 10 pass pass NOUN ejde-39 281 11 theorem theorem NOUN ejde-39 281 12 , , PUNCT ejde-39 281 13 index index NOUN ejde-39 281 14 theory theory NOUN ejde-39 281 15 , , PUNCT ejde-39 281 16 and and CCONJ ejde-39 281 17 ljusternik ljusternik NOUN ejde-39 281 18 - - PUNCT ejde-39 281 19 schnirelman schnirelman NOUN ejde-39 281 20 ’s ’s PART ejde-39 281 21 theory theory NOUN ejde-39 281 22 , , PUNCT ejde-39 281 23 etc etc X ejde-39 281 24 . . X ejde-39 281 25 to to PART ejde-39 281 26 nonlinear nonlinear VERB ejde-39 281 27 generalized generalized ADJ ejde-39 281 28 rayleigh rayleigh PROPN ejde-39 281 29 quotients quotient NOUN ejde-39 281 30 is be AUX ejde-39 281 31 limited limit VERB ejde-39 281 32 in in ADP ejde-39 281 33 light light NOUN ejde-39 281 34 of of ADP ejde-39 281 35 prohibitive prohibitive ADJ ejde-39 281 36 regularity regularity NOUN ejde-39 281 37 and and CCONJ ejde-39 281 38 non non ADJ ejde-39 281 39 - - ADJ ejde-39 281 40 degeneracy degeneracy ADJ ejde-39 281 41 conditions condition NOUN ejde-39 281 42 for for ADP ejde-39 281 43 variational variational ADJ ejde-39 281 44 functionals functional NOUN ejde-39 281 45 . . PUNCT ejde-39 282 1 indeed indeed ADV ejde-39 282 2 , , PUNCT ejde-39 282 3 the the DET ejde-39 282 4 energy energy NOUN ejde-39 282 5 - - PUNCT ejde-39 282 6 level level NOUN ejde-39 282 7 nonlinear nonlinear ADJ ejde-39 282 8 rayleigh rayleigh PROPN ejde-39 282 9 quotient quotient NOUN ejde-39 282 10 reλ reλ NOUN ejde-39 282 11 ( ( PUNCT ejde-39 282 12 u u NOUN ejde-39 282 13 ) ) PUNCT ejde-39 282 14 corresponding correspond VERB ejde-39 282 15 to to ADP ejde-39 282 16 problem problem NOUN ejde-39 282 17 ( ( PUNCT ejde-39 282 18 1.1 1.1 NUM ejde-39 282 19 ) ) PUNCT ejde-39 282 20 is be AUX ejde-39 282 21 not not PART ejde-39 282 22 regular regular ADJ ejde-39 282 23 at at ADP ejde-39 282 24 zero zero NUM ejde-39 282 25 . . PUNCT ejde-39 283 1 hence hence ADV ejde-39 283 2 , , PUNCT ejde-39 283 3 direct direct ADJ ejde-39 283 4 applying apply VERB ejde-39 283 5 the the DET ejde-39 283 6 mountain mountain NOUN ejde-39 283 7 pass pass NOUN ejde-39 283 8 theorem theorem NOUN ejde-39 283 9 in in ADP ejde-39 283 10 this this DET ejde-39 283 11 case case NOUN ejde-39 283 12 is be AUX ejde-39 283 13 impossible impossible ADJ ejde-39 283 14 . . PUNCT ejde-39 284 1 we we PRON ejde-39 284 2 have have AUX ejde-39 284 3 overcome overcome VERB ejde-39 284 4 this this DET ejde-39 284 5 difficulty difficulty NOUN ejde-39 284 6 in in ADP ejde-39 284 7 the the DET ejde-39 284 8 present present ADJ ejde-39 284 9 work work NOUN ejde-39 284 10 by by ADP ejde-39 284 11 introducing introduce VERB ejde-39 284 12 an an DET ejde-39 284 13 appropriate appropriate ADJ ejde-39 284 14 truncation truncation NOUN ejde-39 284 15 function function NOUN ejde-39 284 16 . . PUNCT ejde-39 285 1 we we PRON ejde-39 285 2 believe believe VERB ejde-39 285 3 , , PUNCT ejde-39 285 4 however however ADV ejde-39 285 5 , , PUNCT ejde-39 285 6 there there PRON ejde-39 285 7 are be VERB ejde-39 285 8 other other ADJ ejde-39 285 9 ways way NOUN ejde-39 285 10 for for ADP ejde-39 285 11 overcoming overcome VERB ejde-39 285 12 this this DET ejde-39 285 13 obstacle obstacle NOUN ejde-39 285 14 . . PUNCT ejde-39 286 1 for for ADP ejde-39 286 2 instance instance NOUN ejde-39 286 3 , , PUNCT ejde-39 286 4 one one PRON ejde-39 286 5 might might AUX ejde-39 286 6 try try VERB ejde-39 286 7 to to PART ejde-39 286 8 answer answer VERB ejde-39 286 9 the the DET ejde-39 286 10 question question NOUN ejde-39 286 11 : : PUNCT ejde-39 286 12 is be AUX ejde-39 286 13 it it PRON ejde-39 286 14 possible possible ADJ ejde-39 286 15 to to PART ejde-39 286 16 develop develop VERB ejde-39 286 17 general general ADJ ejde-39 286 18 methods method NOUN ejde-39 286 19 , , PUNCT ejde-39 286 20 like like ADP ejde-39 286 21 mountain mountain NOUN ejde-39 286 22 pass pass NOUN ejde-39 286 23 theorem theorem PROPN ejde-39 286 24 , , PUNCT ejde-39 286 25 etc etc X ejde-39 286 26 , , PUNCT ejde-39 286 27 applicable applicable ADJ ejde-39 286 28 to to ADP ejde-39 286 29 the the DET ejde-39 286 30 rayleigh rayleigh PROPN ejde-39 286 31 quotient quotient NOUN ejde-39 286 32 type type NOUN ejde-39 286 33 function function NOUN ejde-39 286 34 ? ? PUNCT ejde-39 287 1 the the DET ejde-39 287 2 answer answer NOUN ejde-39 287 3 to to ADP ejde-39 287 4 this this DET ejde-39 287 5 question question NOUN ejde-39 287 6 would would AUX ejde-39 287 7 help help VERB ejde-39 287 8 apparently apparently ADV ejde-39 287 9 resolve resolve VERB ejde-39 287 10 a a DET ejde-39 287 11 number number NOUN ejde-39 287 12 of of ADP ejde-39 287 13 open open ADJ ejde-39 287 14 problems problem NOUN ejde-39 287 15 . . PUNCT ejde-39 288 1 10 10 NUM ejde-39 288 2 y. y. PROPN ejde-39 288 3 il’yasov il’yasov PROPN ejde-39 288 4 , , PUNCT ejde-39 288 5 e. e. PROPN ejde-39 288 6 d. d. PROPN ejde-39 288 7 silva silva PROPN ejde-39 288 8 , , PUNCT ejde-39 288 9 m. m. PROPN ejde-39 288 10 l. l. PROPN ejde-39 288 11 silva silva PROPN ejde-39 288 12 ejde-2023/23 ejde-2023/23 PROPN ejde-39 288 13 6 6 NUM ejde-39 288 14 . . PUNCT ejde-39 289 1 appendix appendix NOUN ejde-39 289 2 we we PRON ejde-39 289 3 use use VERB ejde-39 289 4 a a DET ejde-39 289 5 generalized generalized ADJ ejde-39 289 6 version version NOUN ejde-39 289 7 of of ADP ejde-39 289 8 the the DET ejde-39 289 9 benci benci PROPN ejde-39 289 10 - - PUNCT ejde-39 289 11 rabinowitz rabinowitz PROPN ejde-39 289 12 linking linking NOUN ejde-39 289 13 theorem theorem ADJ ejde-39 289 14 [ [ X ejde-39 289 15 5 5 NUM ejde-39 289 16 ] ] PUNCT ejde-39 289 17 for for ADP ejde-39 289 18 functionals functional NOUN ejde-39 289 19 satisfying satisfy VERB ejde-39 289 20 ( ( PUNCT ejde-39 289 21 ce)c ce)c ADJ ejde-39 289 22 condition condition NOUN ejde-39 289 23 . . PUNCT ejde-39 290 1 this this PRON ejde-39 290 2 was be AUX ejde-39 290 3 developed develop VERB ejde-39 290 4 by by ADP ejde-39 290 5 d. d. PROPN ejde-39 290 6 motreanu motreanu PROPN ejde-39 290 7 , , PUNCT ejde-39 290 8 v. v. ADP ejde-39 290 9 motreanu motreanu NOUN ejde-39 290 10 , , PUNCT ejde-39 290 11 n. n. NOUN ejde-39 290 12 papageorgiou papageorgiou NOUN ejde-39 291 1 [ [ X ejde-39 291 2 25 25 NUM ejde-39 291 3 ] ] PUNCT ejde-39 291 4 . . PUNCT ejde-39 292 1 let let AUX ejde-39 292 2 ( ( PUNCT ejde-39 292 3 w w PROPN ejde-39 292 4 , , PUNCT ejde-39 292 5 ‖ ‖ PROPN ejde-39 292 6 · · PUNCT ejde-39 292 7 ‖w ‖w NOUN ejde-39 292 8 ) ) PUNCT ejde-39 292 9 be be AUX ejde-39 292 10 a a DET ejde-39 292 11 banach banach NOUN ejde-39 292 12 space space NOUN ejde-39 292 13 , , PUNCT ejde-39 292 14 b0 b0 VERB ejde-39 292 15 ⊂ ⊂ PROPN ejde-39 292 16 b b PROPN ejde-39 292 17 , , PUNCT ejde-39 293 1 c c AUX ejde-39 293 2 be be VERB ejde-39 293 3 nonempty nonempty X ejde-39 293 4 sets set NOUN ejde-39 293 5 in in ADP ejde-39 293 6 w w NOUN ejde-39 293 7 , , PUNCT ejde-39 293 8 and and CCONJ ejde-39 293 9 idb0 idb0 PROPN ejde-39 293 10 is be AUX ejde-39 293 11 an an DET ejde-39 293 12 identity identity NOUN ejde-39 293 13 map map NOUN ejde-39 293 14 in in ADP ejde-39 293 15 b0 b0 NOUN ejde-39 293 16 . . PUNCT ejde-39 294 1 the the DET ejde-39 294 2 pair pair NOUN ejde-39 294 3 { { PUNCT ejde-39 294 4 b0 b0 NOUN ejde-39 294 5 , , PUNCT ejde-39 294 6 b b NOUN ejde-39 294 7 } } PUNCT ejde-39 294 8 is be AUX ejde-39 294 9 said say VERB ejde-39 294 10 to to PART ejde-39 294 11 be be AUX ejde-39 294 12 links link NOUN ejde-39 294 13 c c NOUN ejde-39 294 14 in in ADP ejde-39 294 15 w w NOUN ejde-39 294 16 if if SCONJ ejde-39 294 17 the the DET ejde-39 294 18 following follow VERB ejde-39 294 19 conditions condition NOUN ejde-39 294 20 hold hold VERB ejde-39 294 21 : : PUNCT ejde-39 294 22 ( ( PUNCT ejde-39 294 23 a a X ejde-39 294 24 ) ) PUNCT ejde-39 294 25 b0 b0 NOUN ejde-39 294 26 ∩ ∩ ADJ ejde-39 294 27 c c NOUN ejde-39 294 28 = = SYM ejde-39 294 29 ∅ ∅ NOUN ejde-39 294 30 ; ; PUNCT ejde-39 294 31 ( ( PUNCT ejde-39 294 32 b b X ejde-39 294 33 ) ) PUNCT ejde-39 294 34 for for ADP ejde-39 294 35 any any DET ejde-39 294 36 h h NOUN ejde-39 294 37 ∈ ∈ PROPN ejde-39 294 38 c(b;w c(b;w PROPN ejde-39 294 39 ) ) PUNCT ejde-39 294 40 with with ADP ejde-39 294 41 h|b0 h|b0 NOUN ejde-39 294 42 = = PUNCT ejde-39 294 43 idb0 idb0 VERB ejde-39 294 44 it it PRON ejde-39 294 45 holds hold VERB ejde-39 294 46 h(b h(b PROPN ejde-39 294 47 ) ) PUNCT ejde-39 294 48 ∩ ∩ PROPN ejde-39 294 49 c c PROPN ejde-39 294 50 6= 6= PROPN ejde-39 294 51 ∅. ∅. ADP ejde-39 294 52 the the DET ejde-39 294 53 following following ADJ ejde-39 294 54 result result NOUN ejde-39 294 55 follows follow VERB ejde-39 294 56 from from ADP ejde-39 294 57 [ [ X ejde-39 294 58 25 25 NUM ejde-39 294 59 , , PUNCT ejde-39 294 60 theorem theorem VERB ejde-39 294 61 5.39 5.39 NUM ejde-39 294 62 ] ] PUNCT ejde-39 294 63 . . PUNCT ejde-39 295 1 theorem theorem NOUN ejde-39 295 2 6.1 6.1 NUM ejde-39 295 3 . . PUNCT ejde-39 296 1 let let AUX ejde-39 296 2 { { PUNCT ejde-39 296 3 b0 b0 VERB ejde-39 296 4 , , PUNCT ejde-39 296 5 b b NOUN ejde-39 296 6 } } PUNCT ejde-39 296 7 links link NOUN ejde-39 296 8 c c NOUN ejde-39 296 9 in in ADP ejde-39 296 10 w w PROPN ejde-39 296 11 , , PUNCT ejde-39 296 12 c c PROPN ejde-39 296 13 closed close VERB ejde-39 296 14 , , PUNCT ejde-39 296 15 d(b0 d(b0 NOUN ejde-39 296 16 , , PUNCT ejde-39 296 17 c c NOUN ejde-39 296 18 ) ) PUNCT ejde-39 296 19 > > X ejde-39 296 20 0 0 X ejde-39 296 21 . . PUNCT ejde-39 297 1 let let VERB ejde-39 297 2 γ γ X ejde-39 297 3 = = PRON ejde-39 297 4 { { PUNCT ejde-39 297 5 h h NOUN ejde-39 297 6 ∈ ∈ PROPN ejde-39 297 7 c(b;w c(b;w PROPN ejde-39 297 8 ) ) PUNCT ejde-39 297 9 : : PUNCT ejde-39 297 10 h|b0 h|b0 X ejde-39 297 11 = = SYM ejde-39 297 12 idb0 idb0 PROPN ejde-39 297 13 } } PUNCT ejde-39 297 14 and and CCONJ ejde-39 297 15 φ φ NUM ejde-39 297 16 ∈ ∈ PROPN ejde-39 297 17 c1(w c1(w NOUN ejde-39 297 18 , , PUNCT ejde-39 297 19 r r NOUN ejde-39 297 20 ) ) PUNCT ejde-39 297 21 be be AUX ejde-39 297 22 such such ADJ ejde-39 297 23 that that DET ejde-39 297 24 b b X ejde-39 297 25 : : PUNCT ejde-39 297 26 = = SYM ejde-39 297 27 supu∈b0 supu∈b0 ADJ ejde-39 297 28 φ(u φ(u NOUN ejde-39 297 29 ) ) PUNCT ejde-39 297 30 ≤ ≤ NUM ejde-39 297 31 infu∈s+ infu∈s+ NOUN ejde-39 297 32 ρ ρ PROPN ejde-39 297 33 φ(u φ(u NOUN ejde-39 297 34 ) ) PUNCT ejde-39 298 1 = = NOUN ejde-39 298 2 : : PUNCT ejde-39 298 3 a. a. NOUN ejde-39 298 4 let let VERB ejde-39 298 5 c c NOUN ejde-39 298 6 : : PUNCT ejde-39 298 7 = = SYM ejde-39 298 8 inf inf PROPN ejde-39 298 9 h∈γ h∈γ NOUN ejde-39 298 10 max max PROPN ejde-39 298 11 u∈b u∈b PROPN ejde-39 298 12 φ(h(u φ(h(u PROPN ejde-39 298 13 ) ) PUNCT ejde-39 298 14 ) ) PUNCT ejde-39 298 15 , , PUNCT ejde-39 298 16 ( ( PUNCT ejde-39 298 17 6.1 6.1 NUM ejde-39 298 18 ) ) PUNCT ejde-39 298 19 and and CCONJ ejde-39 298 20 assume assume VERB ejde-39 298 21 that that SCONJ ejde-39 298 22 φ φ PROPN ejde-39 298 23 satisfying satisfy VERB ejde-39 298 24 the the DET ejde-39 298 25 ( ( PUNCT ejde-39 298 26 ce)-condition ce)-condition NOUN ejde-39 298 27 at at ADP ejde-39 298 28 c. c. PROPN ejde-39 298 29 then then ADV ejde-39 298 30 c c PROPN ejde-39 298 31 ≥ ≥ PROPN ejde-39 298 32 a a PRON ejde-39 298 33 and and CCONJ ejde-39 298 34 c c NOUN ejde-39 298 35 is be AUX ejde-39 298 36 a a DET ejde-39 298 37 critical critical ADJ ejde-39 298 38 value value NOUN ejde-39 298 39 of of ADP ejde-39 298 40 φ φ PROPN ejde-39 298 41 , , PUNCT ejde-39 298 42 i.e. i.e. X ejde-39 298 43 , , PUNCT ejde-39 298 44 there there PRON ejde-39 298 45 exists exist VERB ejde-39 298 46 u u NOUN ejde-39 298 47 ∈w ∈w VERB ejde-39 298 48 \ \ NOUN ejde-39 298 49 0 0 NUM ejde-39 298 50 such such ADJ ejde-39 298 51 that that PRON ejde-39 298 52 dφ(u dφ(u NOUN ejde-39 298 53 ) ) PUNCT ejde-39 298 54 = = SYM ejde-39 298 55 0 0 NUM ejde-39 298 56 and and CCONJ ejde-39 298 57 φ(u φ(u PROPN ejde-39 298 58 ) ) PUNCT ejde-39 298 59 = = SYM ejde-39 298 60 c. c. NOUN ejde-39 298 61 acknowledgments acknowledgment NOUN ejde-39 298 62 . . PUNCT ejde-39 299 1 y. y. PROPN ejde-39 299 2 il’yasov il’yasov PROPN ejde-39 299 3 was be AUX ejde-39 299 4 supported support VERB ejde-39 299 5 by by ADP ejde-39 299 6 rsf rsf PROPN ejde-39 299 7 grant grant NOUN ejde-39 299 8 no no INTJ ejde-39 299 9 . . NOUN ejde-39 300 1 22 22 NUM ejde-39 300 2 - - SYM ejde-39 300 3 21 21 NUM ejde-39 300 4 - - PUNCT ejde-39 300 5 00580 00580 NUM ejde-39 300 6 . . PUNCT ejde-39 301 1 references reference NOUN ejde-39 301 2 [ [ X ejde-39 301 3 1 1 NUM ejde-39 301 4 ] ] PUNCT ejde-39 301 5 a. a. NOUN ejde-39 301 6 ambrosetti ambrosetti PROPN ejde-39 301 7 , , PUNCT ejde-39 301 8 p. p. NOUN ejde-39 301 9 h. h. PROPN ejde-39 301 10 rabinowitz rabinowitz PROPN ejde-39 301 11 ; ; PUNCT ejde-39 301 12 dual dual ADJ ejde-39 301 13 variational variational ADJ ejde-39 301 14 methods method NOUN ejde-39 301 15 in in ADP ejde-39 301 16 critical critical ADJ ejde-39 301 17 point point NOUN ejde-39 301 18 theory theory NOUN ejde-39 301 19 and and CCONJ ejde-39 301 20 applications application NOUN ejde-39 301 21 j. j. PROPN ejde-39 301 22 func func PROPN ejde-39 301 23 . . PUNCT ejde-39 302 1 anal anal PROPN ejde-39 302 2 . . PROPN ejde-39 302 3 , , PUNCT ejde-39 302 4 14 14 NUM ejde-39 302 5 ( ( PUNCT ejde-39 302 6 1973 1973 NUM ejde-39 302 7 ) ) PUNCT ejde-39 302 8 , , PUNCT ejde-39 302 9 349–381 349–381 NUM ejde-39 302 10 . . PUNCT ejde-39 303 1 [ [ X ejde-39 303 2 2 2 X ejde-39 303 3 ] ] PUNCT ejde-39 303 4 s. s. PROPN ejde-39 303 5 n. n. PROPN ejde-39 303 6 antontsev antontsev PROPN ejde-39 303 7 , , PUNCT ejde-39 303 8 j. j. PROPN ejde-39 303 9 i. i. PROPN ejde-39 303 10 dı́az dı́az PROPN ejde-39 303 11 , , PUNCT ejde-39 303 12 s. s. PROPN ejde-39 303 13 shmarev shmarev PROPN ejde-39 303 14 ; ; PUNCT ejde-39 303 15 energy energy NOUN ejde-39 303 16 methods method NOUN ejde-39 303 17 for for ADP ejde-39 303 18 free free ADJ ejde-39 303 19 boundary boundary ADJ ejde-39 303 20 problems problem NOUN ejde-39 303 21 : : PUNCT ejde-39 303 22 applications application NOUN ejde-39 303 23 to to PART ejde-39 303 24 nonlinear nonlinear ADJ ejde-39 303 25 pdes pde NOUN ejde-39 303 26 and and CCONJ ejde-39 303 27 fluid fluid ADJ ejde-39 303 28 mechanics mechanic NOUN ejde-39 303 29 . . PUNCT ejde-39 304 1 vol vol NOUN ejde-39 304 2 . . PROPN ejde-39 305 1 46 46 NUM ejde-39 305 2 , , PUNCT ejde-39 305 3 springer springer NOUN ejde-39 305 4 science science PROPN ejde-39 305 5 & & CCONJ ejde-39 305 6 business business NOUN ejde-39 305 7 media medium NOUN ejde-39 305 8 , , PUNCT ejde-39 305 9 2001 2001 NUM ejde-39 305 10 . . PUNCT ejde-39 306 1 [ [ X ejde-39 306 2 3 3 X ejde-39 306 3 ] ] PUNCT ejde-39 306 4 v. v. PROPN ejde-39 306 5 i. i. PROPN ejde-39 306 6 arnol’d arnol’d PROPN ejde-39 306 7 ; ; PUNCT ejde-39 306 8 catastrophe catastrophe NOUN ejde-39 306 9 theory theory NOUN ejde-39 306 10 , , PUNCT ejde-39 306 11 springer springer NOUN ejde-39 306 12 science science PROPN ejde-39 306 13 & & CCONJ ejde-39 306 14 business business NOUN ejde-39 306 15 media medium NOUN ejde-39 306 16 , , PUNCT ejde-39 306 17 2003 2003 NUM ejde-39 306 18 . . PUNCT ejde-39 307 1 [ [ X ejde-39 307 2 4 4 X ejde-39 307 3 ] ] PUNCT ejde-39 307 4 j. j. PROPN ejde-39 307 5 bellazzini bellazzini PROPN ejde-39 307 6 , , PUNCT ejde-39 307 7 l. l. PROPN ejde-39 307 8 jeanjean jeanjean PROPN ejde-39 307 9 , , PUNCT ejde-39 307 10 t. t. PROPN ejde-39 307 11 luo luo PROPN ejde-39 307 12 ; ; PUNCT ejde-39 307 13 existence existence NOUN ejde-39 307 14 and and CCONJ ejde-39 307 15 instability instability NOUN ejde-39 307 16 of of ADP ejde-39 307 17 standing standing ADJ ejde-39 307 18 waves wave NOUN ejde-39 307 19 with with ADP ejde-39 307 20 prescribed prescribed ADJ ejde-39 307 21 norm norm NOUN ejde-39 307 22 for for ADP ejde-39 307 23 a a DET ejde-39 307 24 class class NOUN ejde-39 307 25 of of ADP ejde-39 307 26 schrödinger schrödinger NOUN ejde-39 307 27 - - PUNCT ejde-39 307 28 poisson poisson NOUN ejde-39 307 29 equations equation NOUN ejde-39 307 30 , , PUNCT ejde-39 307 31 proceedings proceeding NOUN ejde-39 307 32 of of ADP ejde-39 307 33 the the DET ejde-39 307 34 london london PROPN ejde-39 307 35 mathematical mathematical ADJ ejde-39 307 36 society society NOUN ejde-39 307 37 , , PUNCT ejde-39 307 38 107 107 NUM ejde-39 307 39 ( ( PUNCT ejde-39 307 40 2 2 NUM ejde-39 307 41 ) ) PUNCT ejde-39 307 42 ( ( PUNCT ejde-39 307 43 2013 2013 NUM ejde-39 307 44 ) ) PUNCT ejde-39 307 45 , , PUNCT ejde-39 307 46 303–339 303–339 NUM ejde-39 307 47 . . PUNCT ejde-39 308 1 [ [ X ejde-39 308 2 5 5 NUM ejde-39 308 3 ] ] PUNCT ejde-39 308 4 v. v. PROPN ejde-39 308 5 benci benci PROPN ejde-39 308 6 , , PUNCT ejde-39 309 1 p. p. PROPN ejde-39 309 2 h. h. PROPN ejde-39 309 3 rabinowitz rabinowitz PROPN ejde-39 309 4 ; ; PUNCT ejde-39 309 5 critical critical ADJ ejde-39 309 6 point point NOUN ejde-39 309 7 theorems theorem NOUN ejde-39 309 8 for for ADP ejde-39 309 9 indefinite indefinite ADJ ejde-39 309 10 functionals functional NOUN ejde-39 309 11 , , PUNCT ejde-39 309 12 inv inv ADJ ejde-39 309 13 . . PUNCT ejde-39 309 14 math math NOUN ejde-39 309 15 . . PUNCT ejde-39 310 1 , , PUNCT ejde-39 310 2 52 52 NUM ejde-39 310 3 ( ( PUNCT ejde-39 310 4 1979 1979 NUM ejde-39 310 5 ) ) PUNCT ejde-39 310 6 , , PUNCT ejde-39 310 7 241–273 241–273 NUM ejde-39 310 8 . . PUNCT ejde-39 311 1 [ [ X ejde-39 311 2 6 6 NUM ejde-39 311 3 ] ] X ejde-39 311 4 h. h. NOUN ejde-39 311 5 berestycki berestycki PROPN ejde-39 311 6 , , PUNCT ejde-39 311 7 p.-l p.-l NOUN ejde-39 311 8 . . PUNCT ejde-39 312 1 lions lion NOUN ejde-39 312 2 ; ; PUNCT ejde-39 312 3 nonlinear nonlinear ADJ ejde-39 312 4 scalar scalar ADJ ejde-39 312 5 field field NOUN ejde-39 312 6 equations equation NOUN ejde-39 312 7 , , PUNCT ejde-39 312 8 pt pt X ejde-39 312 9 . . PROPN ejde-39 312 10 1 1 NUM ejde-39 312 11 , , PUNCT ejde-39 312 12 archive archive NOUN ejde-39 312 13 for for ADP ejde-39 312 14 rational rational ADJ ejde-39 312 15 mechanics mechanic NOUN ejde-39 312 16 and and CCONJ ejde-39 312 17 analysis analysis NOUN ejde-39 312 18 , , PUNCT ejde-39 312 19 82 82 NUM ejde-39 312 20 , , PUNCT ejde-39 312 21 ( ( PUNCT ejde-39 312 22 4 4 NUM ejde-39 312 23 ) ) PUNCT ejde-39 312 24 ( ( PUNCT ejde-39 312 25 1983 1983 NUM ejde-39 312 26 ) ) PUNCT ejde-39 312 27 , , PUNCT ejde-39 312 28 313–346 313–346 NUM ejde-39 312 29 . . PUNCT ejde-39 313 1 [ [ X ejde-39 313 2 7 7 X ejde-39 313 3 ] ] X ejde-39 313 4 r. r. PROPN ejde-39 313 5 carles carles PROPN ejde-39 313 6 , , PUNCT ejde-39 313 7 y. y. PROPN ejde-39 313 8 il’yasov il’yasov PROPN ejde-39 313 9 ; ; PUNCT ejde-39 313 10 on on ADP ejde-39 313 11 ground ground NOUN ejde-39 313 12 states state NOUN ejde-39 313 13 for for ADP ejde-39 313 14 the the DET ejde-39 313 15 2d 2d PROPN ejde-39 313 16 schrödinger schrödinger NOUN ejde-39 313 17 equation equation NOUN ejde-39 313 18 with with ADP ejde-39 313 19 combined combined ADJ ejde-39 313 20 nonlinearities nonlinearitie NOUN ejde-39 313 21 and and CCONJ ejde-39 313 22 harmonic harmonic ADJ ejde-39 313 23 potential potential NOUN ejde-39 313 24 , , PUNCT ejde-39 313 25 studies study NOUN ejde-39 313 26 in in ADP ejde-39 313 27 appl appl PROPN ejde-39 313 28 . . PUNCT ejde-39 313 29 math math PROPN ejde-39 313 30 . . PUNCT ejde-39 313 31 , , PUNCT ejde-39 313 32 150 150 NUM ejde-39 313 33 ( ( PUNCT ejde-39 313 34 1 1 NUM ejde-39 313 35 ) ) PUNCT ejde-39 313 36 ( ( PUNCT ejde-39 313 37 2023 2023 NUM ejde-39 313 38 ) ) PUNCT ejde-39 313 39 92–118 92–118 NUM ejde-39 313 40 . . PUNCT ejde-39 314 1 [ [ X ejde-39 314 2 8 8 NUM ejde-39 314 3 ] ] PUNCT ejde-39 314 4 m. m. NOUN ejde-39 314 5 l. l. PROPN ejde-39 314 6 carvalho carvalho PROPN ejde-39 314 7 , , PUNCT ejde-39 314 8 y. y. PROPN ejde-39 314 9 il’yasov il’yasov PROPN ejde-39 314 10 , , PUNCT ejde-39 314 11 c. c. PROPN ejde-39 314 12 a. a. PROPN ejde-39 314 13 santos santos PROPN ejde-39 314 14 ; ; PUNCT ejde-39 314 15 separating separate VERB ejde-39 314 16 solutions solution NOUN ejde-39 314 17 of of ADP ejde-39 314 18 nonlinear nonlinear ADJ ejde-39 314 19 problems problem NOUN ejde-39 314 20 using use VERB ejde-39 314 21 nonlinear nonlinear ADJ ejde-39 314 22 generalized generalize VERB ejde-39 314 23 rayleigh rayleigh PROPN ejde-39 314 24 quotients quotient NOUN ejde-39 314 25 , , PUNCT ejde-39 314 26 topol topol NOUN ejde-39 314 27 . . PUNCT ejde-39 315 1 methods method NOUN ejde-39 315 2 nonl nonl PROPN ejde-39 315 3 . . PUNCT ejde-39 316 1 anal anal PROPN ejde-39 316 2 . . PROPN ejde-39 316 3 , , PUNCT ejde-39 316 4 58 58 NUM ejde-39 316 5 ( ( PUNCT ejde-39 316 6 2 2 NUM ejde-39 316 7 ) ) PUNCT ejde-39 316 8 ( ( PUNCT ejde-39 316 9 2021 2021 NUM ejde-39 316 10 ) ) PUNCT ejde-39 316 11 , , PUNCT ejde-39 316 12 453 453 NUM ejde-39 316 13 – – PUNCT ejde-39 316 14 480 480 NUM ejde-39 316 15 . . PUNCT ejde-39 317 1 [ [ X ejde-39 317 2 9 9 NUM ejde-39 317 3 ] ] PUNCT ejde-39 317 4 m. m. NOUN ejde-39 317 5 l. l. PROPN ejde-39 317 6 carvalho carvalho PROPN ejde-39 317 7 , , PUNCT ejde-39 317 8 y. y. PROPN ejde-39 317 9 il’yasov il’yasov PROPN ejde-39 317 10 , , PUNCT ejde-39 317 11 c. c. PROPN ejde-39 317 12 a. a. PROPN ejde-39 317 13 santos santos PROPN ejde-39 317 14 ; ; PUNCT ejde-39 317 15 existence existence NOUN ejde-39 317 16 of of ADP ejde-39 317 17 s s NOUN ejde-39 317 18 - - PUNCT ejde-39 317 19 shaped shape VERB ejde-39 317 20 type type NOUN ejde-39 317 21 bifurcation bifurcation NOUN ejde-39 317 22 curve curve NOUN ejde-39 317 23 with with ADP ejde-39 317 24 dual dual ADJ ejde-39 317 25 cusp cusp NOUN ejde-39 317 26 catastrophe catastrophe NOUN ejde-39 317 27 via via ADP ejde-39 317 28 variational variational ADJ ejde-39 317 29 methods method NOUN ejde-39 317 30 . . PUNCT ejde-39 318 1 j. j. PROPN ejde-39 318 2 diff diff PROPN ejde-39 318 3 . . PUNCT ejde-39 319 1 eq eq ADP ejde-39 319 2 . . PROPN ejde-39 319 3 , , PUNCT ejde-39 319 4 334 334 NUM ejde-39 319 5 ( ( PUNCT ejde-39 319 6 2022 2022 NUM ejde-39 319 7 ) ) PUNCT ejde-39 319 8 , , PUNCT ejde-39 319 9 256 256 NUM ejde-39 319 10 - - SYM ejde-39 319 11 279 279 NUM ejde-39 319 12 . . PUNCT ejde-39 320 1 [ [ X ejde-39 320 2 10 10 NUM ejde-39 320 3 ] ] PUNCT ejde-39 320 4 t. t. PROPN ejde-39 320 5 cazenave cazenave PROPN ejde-39 320 6 ; ; PUNCT ejde-39 320 7 semilinear semilinear PROPN ejde-39 320 8 schrödinger schrödinger PROPN ejde-39 320 9 equations equation NOUN ejde-39 320 10 , , PUNCT ejde-39 320 11 courant courant ADJ ejde-39 320 12 lecture lecture NOUN ejde-39 320 13 notes note NOUN ejde-39 320 14 in in ADP ejde-39 320 15 mathematics mathematic NOUN ejde-39 320 16 , , PUNCT ejde-39 320 17 10 10 NUM ejde-39 320 18 , , PUNCT ejde-39 320 19 2003 2003 NUM ejde-39 320 20 . . PUNCT ejde-39 321 1 [ [ X ejde-39 321 2 11 11 NUM ejde-39 321 3 ] ] PUNCT ejde-39 321 4 t. t. PROPN ejde-39 321 5 cazenave cazenave PROPN ejde-39 321 6 , , PUNCT ejde-39 321 7 p. p. NOUN ejde-39 321 8 l. l. PROPN ejde-39 321 9 lions lion NOUN ejde-39 321 10 ; ; PUNCT ejde-39 321 11 orbital orbital ADJ ejde-39 321 12 stability stability NOUN ejde-39 321 13 of of ADP ejde-39 321 14 standing stand VERB ejde-39 321 15 waves wave NOUN ejde-39 321 16 for for ADP ejde-39 321 17 some some DET ejde-39 321 18 nonlinear nonlinear ADJ ejde-39 321 19 schrödinger schrödinger NOUN ejde-39 321 20 equations equation NOUN ejde-39 321 21 , , PUNCT ejde-39 321 22 communications communication NOUN ejde-39 321 23 in in ADP ejde-39 321 24 mathematical mathematical ADJ ejde-39 321 25 physics physics NOUN ejde-39 321 26 , , PUNCT ejde-39 321 27 85 85 NUM ejde-39 321 28 ( ( PUNCT ejde-39 321 29 4 4 NUM ejde-39 321 30 ) ) PUNCT ejde-39 321 31 ( ( PUNCT ejde-39 321 32 1982 1982 NUM ejde-39 321 33 ) ) PUNCT ejde-39 321 34 , , PUNCT ejde-39 321 35 549–561 549–561 NUM ejde-39 321 36 . . PUNCT ejde-39 322 1 [ [ X ejde-39 322 2 12 12 NUM ejde-39 322 3 ] ] X ejde-39 322 4 d. d. PROPN ejde-39 322 5 colton colton PROPN ejde-39 322 6 , , PUNCT ejde-39 322 7 h. h. PROPN ejde-39 322 8 w. w. PROPN ejde-39 322 9 engl engl PROPN ejde-39 322 10 , , PUNCT ejde-39 322 11 a. a. PROPN ejde-39 322 12 k. k. PROPN ejde-39 322 13 louis louis PROPN ejde-39 322 14 , , PUNCT ejde-39 322 15 j. j. PROPN ejde-39 322 16 mclaughlin mclaughlin PROPN ejde-39 322 17 , , PUNCT ejde-39 322 18 w. w. PROPN ejde-39 322 19 rundell rundell NOUN ejde-39 322 20 ; ; PUNCT ejde-39 322 21 surveys survey NOUN ejde-39 322 22 on on ADP ejde-39 322 23 solution solution NOUN ejde-39 322 24 methods method NOUN ejde-39 322 25 for for ADP ejde-39 322 26 inverse inverse NOUN ejde-39 322 27 problems problem NOUN ejde-39 322 28 . . PUNCT ejde-39 323 1 springer springer NOUN ejde-39 323 2 science science PROPN ejde-39 323 3 & & CCONJ ejde-39 323 4 business business NOUN ejde-39 323 5 media medium NOUN ejde-39 323 6 , , PUNCT ejde-39 323 7 2012 2012 NUM ejde-39 323 8 . . PUNCT ejde-39 324 1 [ [ X ejde-39 324 2 13 13 NUM ejde-39 324 3 ] ] PUNCT ejde-39 324 4 p. p. NOUN ejde-39 324 5 drábek drábek PROPN ejde-39 324 6 , , PUNCT ejde-39 324 7 j. j. PROPN ejde-39 324 8 milota milota PROPN ejde-39 324 9 ; ; PUNCT ejde-39 324 10 methods method NOUN ejde-39 324 11 of of ADP ejde-39 324 12 nonlinear nonlinear ADJ ejde-39 324 13 analysis analysis NOUN ejde-39 324 14 , , PUNCT ejde-39 324 15 applications application NOUN ejde-39 324 16 to to PART ejde-39 324 17 differential differential VERB ejde-39 324 18 equations equation NOUN ejde-39 324 19 . . PUNCT ejde-39 325 1 second second ADJ ejde-39 325 2 edition edition NOUN ejde-39 325 3 . . PUNCT ejde-39 326 1 birkhäuser birkhäuser X ejde-39 326 2 advanced advanced ADJ ejde-39 326 3 texts text NOUN ejde-39 326 4 : : PUNCT ejde-39 326 5 basler basler NOUN ejde-39 326 6 lehrbücher lehrbücher ADP ejde-39 326 7 , , PUNCT ejde-39 326 8 2013 2013 NUM ejde-39 326 9 . . PUNCT ejde-39 327 1 [ [ X ejde-39 327 2 14 14 NUM ejde-39 327 3 ] ] X ejde-39 327 4 d. d. PROPN ejde-39 327 5 e. e. PROPN ejde-39 327 6 edmunds edmunds PROPN ejde-39 327 7 , , PUNCT ejde-39 327 8 w. w. PROPN ejde-39 327 9 d. d. PROPN ejde-39 327 10 evans evans PROPN ejde-39 327 11 ; ; PUNCT ejde-39 327 12 spectral spectral ADJ ejde-39 327 13 theory theory NOUN ejde-39 327 14 and and CCONJ ejde-39 327 15 differential differential ADJ ejde-39 327 16 operators operator NOUN ejde-39 327 17 . . PUNCT ejde-39 328 1 vol vol NOUN ejde-39 328 2 . . PROPN ejde-39 329 1 15 15 NUM ejde-39 329 2 , , PUNCT ejde-39 330 1 oxford oxford NOUN ejde-39 330 2 : : PUNCT ejde-39 330 3 clarendon clarendon PROPN ejde-39 330 4 press press PROPN ejde-39 330 5 , , PUNCT ejde-39 330 6 1987 1987 NUM ejde-39 330 7 . . PUNCT ejde-39 331 1 [ [ X ejde-39 331 2 15 15 NUM ejde-39 331 3 ] ] X ejde-39 331 4 p. p. PROPN ejde-39 331 5 l. l. PROPN ejde-39 331 6 felmer felmer PROPN ejde-39 331 7 ; ; PUNCT ejde-39 331 8 periodic periodic ADJ ejde-39 331 9 - - PUNCT ejde-39 331 10 solutions solution NOUN ejde-39 331 11 of of ADP ejde-39 331 12 “ " PUNCT ejde-39 331 13 superquadratic superquadratic ADJ ejde-39 331 14 ” " PUNCT ejde-39 331 15 hamiltonian hamiltonian ADJ ejde-39 331 16 systems system NOUN ejde-39 331 17 , , PUNCT ejde-39 331 18 journal journal NOUN ejde-39 331 19 of of ADP ejde-39 331 20 differential differential ADJ ejde-39 331 21 equations equation NOUN ejde-39 331 22 , , PUNCT ejde-39 331 23 102 102 NUM ejde-39 331 24 ( ( PUNCT ejde-39 331 25 1 1 NUM ejde-39 331 26 ) ) PUNCT ejde-39 331 27 ( ( PUNCT ejde-39 331 28 1993 1993 NUM ejde-39 331 29 ) ) PUNCT ejde-39 331 30 , , PUNCT ejde-39 331 31 188–207 188–207 NUM ejde-39 331 32 . . PUNCT ejde-39 332 1 [ [ X ejde-39 332 2 16 16 NUM ejde-39 332 3 ] ] X ejde-39 332 4 d. d. PROPN ejde-39 332 5 gilbarg gilbarg PROPN ejde-39 332 6 , , PUNCT ejde-39 332 7 n. n. PROPN ejde-39 332 8 s. s. PROPN ejde-39 332 9 trudinger trudinger PROPN ejde-39 332 10 ; ; PUNCT ejde-39 332 11 elliptic elliptic ADJ ejde-39 332 12 partial partial ADJ ejde-39 332 13 differential differential ADJ ejde-39 332 14 equations equation NOUN ejde-39 332 15 of of ADP ejde-39 332 16 second second ADJ ejde-39 332 17 order order NOUN ejde-39 332 18 , , PUNCT ejde-39 332 19 berlin berlin PROPN ejde-39 332 20 : : PUNCT ejde-39 332 21 springer springer NOUN ejde-39 332 22 , , PUNCT ejde-39 332 23 1977 1977 NUM ejde-39 332 24 . . PUNCT ejde-39 333 1 ejde-2023/23 ejde-2023/23 NOUN ejde-39 333 2 prescribed prescribe VERB ejde-39 333 3 energy energy NOUN ejde-39 333 4 saddle saddle NOUN ejde-39 333 5 - - PUNCT ejde-39 333 6 point point NOUN ejde-39 333 7 solutions solution NOUN ejde-39 333 8 11 11 NUM ejde-39 333 9 [ [ X ejde-39 333 10 17 17 NUM ejde-39 333 11 ] ] X ejde-39 333 12 y. y. PROPN ejde-39 333 13 il’yasov il’yasov PROPN ejde-39 333 14 ; ; PUNCT ejde-39 333 15 rayleigh rayleigh PROPN ejde-39 333 16 quotients quotient NOUN ejde-39 333 17 of of ADP ejde-39 333 18 the the DET ejde-39 333 19 level level NOUN ejde-39 333 20 set set NOUN ejde-39 333 21 manifolds manifold NOUN ejde-39 333 22 related relate VERB ejde-39 333 23 to to ADP ejde-39 333 24 the the DET ejde-39 333 25 nonlinear nonlinear ADJ ejde-39 333 26 pde pde NOUN ejde-39 333 27 , , PUNCT ejde-39 333 28 minimax minimax NOUN ejde-39 333 29 theory theory NOUN ejde-39 333 30 and and CCONJ ejde-39 333 31 its its PRON ejde-39 333 32 applications application NOUN ejde-39 333 33 , , PUNCT ejde-39 333 34 07 07 NUM ejde-39 333 35 ( ( PUNCT ejde-39 333 36 2 2 NUM ejde-39 333 37 ) ) PUNCT ejde-39 333 38 ( ( PUNCT ejde-39 333 39 2022 2022 NUM ejde-39 333 40 ) ) PUNCT ejde-39 333 41 , , PUNCT ejde-39 333 42 277–302 277–302 NUM ejde-39 333 43 . . PUNCT ejde-39 334 1 [ [ X ejde-39 334 2 18 18 NUM ejde-39 334 3 ] ] X ejde-39 334 4 y. y. PROPN ejde-39 334 5 il’yasov il’yasov PROPN ejde-39 334 6 , , PUNCT ejde-39 334 7 a. a. PROPN ejde-39 334 8 b. b. PROPN ejde-39 334 9 muravnik muravnik PROPN ejde-39 334 10 ; ; PUNCT ejde-39 334 11 min min ADJ ejde-39 334 12 - - ADJ ejde-39 334 13 max max PROPN ejde-39 334 14 principles principle NOUN ejde-39 334 15 with with ADP ejde-39 334 16 nonlinear nonlinear ADJ ejde-39 334 17 generalized generalize VERB ejde-39 334 18 rayleigh rayleigh PROPN ejde-39 334 19 quotients quotient NOUN ejde-39 334 20 for for ADP ejde-39 334 21 nonlinear nonlinear ADJ ejde-39 334 22 equations equation NOUN ejde-39 334 23 . . PUNCT ejde-39 335 1 journal journal PROPN ejde-39 335 2 of of ADP ejde-39 335 3 mathematical mathematical ADJ ejde-39 335 4 sciences sciences PROPN ejde-39 335 5 , , PUNCT ejde-39 335 6 260(6 260(6 NUM ejde-39 335 7 ) ) PUNCT ejde-39 335 8 ( ( PUNCT ejde-39 335 9 2022 2022 NUM ejde-39 335 10 ) ) PUNCT ejde-39 335 11 , , PUNCT ejde-39 335 12 738–747 738–747 NUM ejde-39 335 13 . . PUNCT ejde-39 336 1 [ [ X ejde-39 336 2 19 19 NUM ejde-39 336 3 ] ] X ejde-39 336 4 y. y. PROPN ejde-39 336 5 il’yasov il’yasov PROPN ejde-39 336 6 ; ; PUNCT ejde-39 336 7 on on ADP ejde-39 336 8 fundamental fundamental ADJ ejde-39 336 9 frequency frequency NOUN ejde-39 336 10 solutions solution NOUN ejde-39 336 11 with with ADP ejde-39 336 12 prescribed prescribed ADJ ejde-39 336 13 action action NOUN ejde-39 336 14 value value NOUN ejde-39 336 15 of of ADP ejde-39 336 16 the the DET ejde-39 336 17 nls nls NOUN ejde-39 336 18 equations equation NOUN ejde-39 336 19 . . PUNCT ejde-39 337 1 j. j. PROPN ejde-39 337 2 math math PROPN ejde-39 337 3 . . PUNCT ejde-39 338 1 sc sc PROPN ejde-39 338 2 . . PROPN ejde-39 338 3 , , PUNCT ejde-39 338 4 259 259 NUM ejde-39 338 5 ( ( PUNCT ejde-39 338 6 2 2 NUM ejde-39 338 7 ) ) PUNCT ejde-39 338 8 ( ( PUNCT ejde-39 338 9 2021 2021 NUM ejde-39 338 10 ) ) PUNCT ejde-39 338 11 , , PUNCT ejde-39 338 12 187–204 187–204 NUM ejde-39 338 13 . . PUNCT ejde-39 339 1 [ [ X ejde-39 339 2 20 20 NUM ejde-39 339 3 ] ] X ejde-39 339 4 y. y. PROPN ejde-39 339 5 sh sh PROPN ejde-39 339 6 . . PROPN ejde-39 339 7 ilyasov ilyasov PROPN ejde-39 339 8 , , PUNCT ejde-39 339 9 n. n. PROPN ejde-39 339 10 f. f. PROPN ejde-39 339 11 valeev valeev PROPN ejde-39 339 12 ; ; PUNCT ejde-39 339 13 on on ADP ejde-39 339 14 nonlinear nonlinear ADJ ejde-39 339 15 boundary boundary ADJ ejde-39 339 16 value value NOUN ejde-39 339 17 problem problem NOUN ejde-39 339 18 corresponding correspond VERB ejde-39 339 19 to to ADP ejde-39 339 20 ndimensional ndimensional ADJ ejde-39 339 21 inverse inverse ADJ ejde-39 339 22 spectral spectral ADJ ejde-39 339 23 problem problem NOUN ejde-39 339 24 , , PUNCT ejde-39 339 25 j. j. PROPN ejde-39 339 26 diff diff PROPN ejde-39 339 27 . . PUNCT ejde-39 340 1 eq eq ADP ejde-39 340 2 . . PROPN ejde-39 340 3 , , PUNCT ejde-39 340 4 266 266 NUM ejde-39 340 5 ( ( PUNCT ejde-39 340 6 8) 8) NUM ejde-39 340 7 ( ( PUNCT ejde-39 340 8 2019 2019 NUM ejde-39 340 9 ) ) PUNCT ejde-39 340 10 , , PUNCT ejde-39 340 11 4533–4543 4533–4543 NUM ejde-39 340 12 . . PUNCT ejde-39 341 1 [ [ X ejde-39 341 2 21 21 NUM ejde-39 341 3 ] ] X ejde-39 341 4 y. y. NOUN ejde-39 341 5 ilyasov ilyasov NOUN ejde-39 341 6 ; ; PUNCT ejde-39 341 7 on on ADP ejde-39 341 8 extreme extreme ADJ ejde-39 341 9 values value NOUN ejde-39 341 10 of of ADP ejde-39 341 11 nehari nehari PROPN ejde-39 341 12 manifold manifold ADJ ejde-39 341 13 method method NOUN ejde-39 341 14 via via ADP ejde-39 341 15 nonlinear nonlinear PROPN ejde-39 341 16 rayleigh rayleigh PROPN ejde-39 341 17 ’s ’s PART ejde-39 341 18 quotient quotient NOUN ejde-39 341 19 , , PUNCT ejde-39 341 20 tmna tmna PROPN ejde-39 341 21 , , PUNCT ejde-39 341 22 49 49 NUM ejde-39 341 23 ( ( PUNCT ejde-39 341 24 2 2 NUM ejde-39 341 25 ) ) PUNCT ejde-39 341 26 ( ( PUNCT ejde-39 341 27 2017 2017 NUM ejde-39 341 28 ) ) PUNCT ejde-39 341 29 , , PUNCT ejde-39 341 30 683–714 683–714 NUM ejde-39 341 31 . . PUNCT ejde-39 342 1 [ [ X ejde-39 342 2 22 22 NUM ejde-39 342 3 ] ] PUNCT ejde-39 342 4 l. l. PROPN ejde-39 342 5 jeanjean jeanjean PROPN ejde-39 342 6 ; ; PUNCT ejde-39 342 7 existence existence NOUN ejde-39 342 8 of of ADP ejde-39 342 9 solutions solution NOUN ejde-39 342 10 with with ADP ejde-39 342 11 prescribed prescribed ADJ ejde-39 342 12 norm norm NOUN ejde-39 342 13 for for ADP ejde-39 342 14 semilinear semilinear PROPN ejde-39 342 15 elliptic elliptic ADJ ejde-39 342 16 equations equation NOUN ejde-39 342 17 , , PUNCT ejde-39 342 18 nonlinear nonlinear ADJ ejde-39 342 19 analysis analysis NOUN ejde-39 342 20 : : PUNCT ejde-39 342 21 theory theory NOUN ejde-39 342 22 , , PUNCT ejde-39 342 23 methods method NOUN ejde-39 342 24 & & CCONJ ejde-39 342 25 applications application NOUN ejde-39 342 26 , , PUNCT ejde-39 342 27 28(10 28(10 NUM ejde-39 342 28 ) ) PUNCT ejde-39 342 29 ( ( PUNCT ejde-39 342 30 1997 1997 NUM ejde-39 342 31 ) ) PUNCT ejde-39 342 32 , , PUNCT ejde-39 342 33 1633–1659 1633–1659 NUM ejde-39 342 34 . . PUNCT ejde-39 343 1 [ [ X ejde-39 343 2 23 23 NUM ejde-39 343 3 ] ] X ejde-39 343 4 f. f. PROPN ejde-39 343 5 r. r. PROPN ejde-39 343 6 klinkhamer klinkhamer PROPN ejde-39 343 7 , , PUNCT ejde-39 343 8 s. s. PROPN ejde-39 343 9 m. m. PROPN ejde-39 343 10 nicholas nicholas PROPN ejde-39 343 11 ; ; PUNCT ejde-39 343 12 a a DET ejde-39 343 13 saddle saddle NOUN ejde-39 343 14 - - PUNCT ejde-39 343 15 point point NOUN ejde-39 343 16 solution solution NOUN ejde-39 343 17 in in ADP ejde-39 343 18 the the DET ejde-39 343 19 weinberg weinberg PROPN ejde-39 343 20 - - PUNCT ejde-39 343 21 salam salam PROPN ejde-39 343 22 theory theory PROPN ejde-39 343 23 , , PUNCT ejde-39 343 24 physical physical ADJ ejde-39 343 25 review review NOUN ejde-39 343 26 d d PROPN ejde-39 343 27 , , PUNCT ejde-39 343 28 30 30 NUM ejde-39 343 29 ( ( PUNCT ejde-39 343 30 1984 1984 NUM ejde-39 343 31 ) ) PUNCT ejde-39 343 32 2212 2212 NUM ejde-39 343 33 . . PUNCT ejde-39 344 1 [ [ X ejde-39 344 2 24 24 NUM ejde-39 344 3 ] ] X ejde-39 344 4 n. n. PROPN ejde-39 344 5 manton manton PROPN ejde-39 344 6 , , PUNCT ejde-39 344 7 p. p. PROPN ejde-39 344 8 sutcliffe sutcliffe PROPN ejde-39 344 9 ; ; PUNCT ejde-39 344 10 topological topological ADJ ejde-39 344 11 solitons soliton NOUN ejde-39 344 12 , , PUNCT ejde-39 344 13 cambridge cambridge PROPN ejde-39 344 14 university university PROPN ejde-39 344 15 press press NOUN ejde-39 344 16 , , PUNCT ejde-39 344 17 2004 2004 NUM ejde-39 344 18 . . PUNCT ejde-39 345 1 [ [ X ejde-39 345 2 25 25 NUM ejde-39 345 3 ] ] X ejde-39 345 4 d. d. PROPN ejde-39 345 5 motreanu motreanu PROPN ejde-39 345 6 , , PUNCT ejde-39 345 7 v. v. PROPN ejde-39 345 8 v. v. CCONJ ejde-39 345 9 motreanu motreanu NOUN ejde-39 345 10 , , PUNCT ejde-39 345 11 n. n. NOUN ejde-39 345 12 papageorgiou papageorgiou NOUN ejde-39 345 13 ; ; PUNCT ejde-39 345 14 topological topological ADJ ejde-39 345 15 and and CCONJ ejde-39 345 16 variational variational ADJ ejde-39 345 17 methods method NOUN ejde-39 345 18 with with ADP ejde-39 345 19 applications application NOUN ejde-39 345 20 to to ADP ejde-39 345 21 nonlinear nonlinear ADJ ejde-39 345 22 boundary boundary ADJ ejde-39 345 23 value value NOUN ejde-39 345 24 problems problem NOUN ejde-39 345 25 , , PUNCT ejde-39 345 26 springer springer NOUN ejde-39 345 27 , , PUNCT ejde-39 345 28 new new PROPN ejde-39 345 29 york york PROPN ejde-39 345 30 , , PUNCT ejde-39 345 31 2014 2014 NUM ejde-39 345 32 . . PUNCT ejde-39 346 1 [ [ X ejde-39 346 2 26 26 NUM ejde-39 346 3 ] ] PUNCT ejde-39 346 4 m. m. NOUN ejde-39 346 5 reed reed PROPN ejde-39 346 6 , , PUNCT ejde-39 346 7 b. b. PROPN ejde-39 346 8 simon simon PROPN ejde-39 346 9 ; ; PUNCT ejde-39 346 10 methods method NOUN ejde-39 346 11 of of ADP ejde-39 346 12 modern modern ADJ ejde-39 346 13 mathematical mathematical ADJ ejde-39 346 14 physics physics NOUN ejde-39 346 15 . . PUNCT ejde-39 347 1 iv iv X ejde-39 347 2 . . PUNCT ejde-39 348 1 analysis analysis NOUN ejde-39 348 2 of of ADP ejde-39 348 3 operators operator NOUN ejde-39 348 4 , , PUNCT ejde-39 348 5 academic academic ADJ ejde-39 348 6 press press NOUN ejde-39 349 1 [ [ X ejde-39 349 2 harcourt harcourt PROPN ejde-39 349 3 brace brace PROPN ejde-39 349 4 jovanovich jovanovich NOUN ejde-39 349 5 , , PUNCT ejde-39 349 6 publishers publisher NOUN ejde-39 349 7 ] ] X ejde-39 349 8 , , PUNCT ejde-39 349 9 1978 1978 NUM ejde-39 349 10 . . PUNCT ejde-39 350 1 [ [ X ejde-39 350 2 27 27 NUM ejde-39 350 3 ] ] X ejde-39 350 4 n. n. PROPN ejde-39 350 5 soave soave PROPN ejde-39 350 6 ; ; PUNCT ejde-39 350 7 normalized normalize VERB ejde-39 350 8 ground ground NOUN ejde-39 350 9 states state NOUN ejde-39 350 10 for for ADP ejde-39 350 11 the the DET ejde-39 350 12 nls nls NOUN ejde-39 350 13 equation equation NOUN ejde-39 350 14 with with ADP ejde-39 350 15 combined combined ADJ ejde-39 350 16 nonlinearities nonlinearitie NOUN ejde-39 350 17 , , PUNCT ejde-39 350 18 j. j. PROPN ejde-39 350 19 diff diff PROPN ejde-39 350 20 . . PUNCT ejde-39 351 1 eqs eqs PROPN ejde-39 351 2 , , PUNCT ejde-39 351 3 269(9)(2020 269(9)(2020 NUM ejde-39 351 4 ) ) PUNCT ejde-39 351 5 6941–6987 6941–6987 NUM ejde-39 351 6 . . PUNCT ejde-39 352 1 [ [ X ejde-39 352 2 28 28 NUM ejde-39 352 3 ] ] X ejde-39 352 4 m. m. NOUN ejde-39 352 5 struwe struwe PROPN ejde-39 352 6 ; ; PUNCT ejde-39 352 7 variational variational ADJ ejde-39 352 8 methods method NOUN ejde-39 352 9 : : PUNCT ejde-39 352 10 applications application NOUN ejde-39 352 11 to to PART ejde-39 352 12 nonlinear nonlinear VERB ejde-39 352 13 partial partial ADJ ejde-39 352 14 differential differential NOUN ejde-39 352 15 equations equation NOUN ejde-39 352 16 and and CCONJ ejde-39 352 17 hamiltonian hamiltonian ADJ ejde-39 352 18 systems system NOUN ejde-39 352 19 , , PUNCT ejde-39 352 20 springer springer NOUN ejde-39 352 21 , , PUNCT ejde-39 352 22 2008 2008 NUM ejde-39 352 23 . . PUNCT ejde-39 353 1 [ [ X ejde-39 353 2 29 29 NUM ejde-39 353 3 ] ] X ejde-39 353 4 b. b. PROPN ejde-39 353 5 n. n. PROPN ejde-39 353 6 zakhariev zakhariev PROPN ejde-39 353 7 , , PUNCT ejde-39 353 8 a. a. PROPN ejde-39 353 9 a. a. PROPN ejde-39 353 10 suzko suzko PROPN ejde-39 353 11 ; ; PUNCT ejde-39 353 12 direct direct ADJ ejde-39 353 13 and and CCONJ ejde-39 353 14 inverse inverse NOUN ejde-39 353 15 problems problem NOUN ejde-39 353 16 : : PUNCT ejde-39 353 17 potentials potential VERB ejde-39 353 18 in in ADP ejde-39 353 19 quantum quantum ADJ ejde-39 353 20 scattering scattering NOUN ejde-39 353 21 , , PUNCT ejde-39 353 22 springer springer NOUN ejde-39 353 23 science science PROPN ejde-39 353 24 & & CCONJ ejde-39 353 25 business business NOUN ejde-39 353 26 media medium NOUN ejde-39 353 27 , , PUNCT ejde-39 353 28 2012 2012 NUM ejde-39 353 29 . . PUNCT ejde-39 354 1 yavdat yavdat PROPN ejde-39 354 2 il’yasov il’yasov PROPN ejde-39 354 3 institute institute PROPN ejde-39 354 4 of of ADP ejde-39 354 5 mathematics mathematics PROPN ejde-39 354 6 with with ADP ejde-39 354 7 computing computing NOUN ejde-39 354 8 centre centre PROPN ejde-39 354 9 of of ADP ejde-39 354 10 ufa ufa PROPN ejde-39 354 11 , , PUNCT ejde-39 354 12 federal federal ADJ ejde-39 354 13 research research NOUN ejde-39 354 14 centre centre PROPN ejde-39 354 15 , , PUNCT ejde-39 354 16 ras ras PROPN ejde-39 354 17 , , PUNCT ejde-39 354 18 ufa ufa PROPN ejde-39 354 19 , , PUNCT ejde-39 354 20 russia russia PROPN ejde-39 354 21 email email NOUN ejde-39 354 22 address address NOUN ejde-39 354 23 : : PUNCT ejde-39 354 24 ilyasov02@gmail.com ilyasov02@gmail.com PROPN ejde-39 354 25 edcarlos edcarlo VERB ejde-39 354 26 domingos domingos PROPN ejde-39 354 27 da da PROPN ejde-39 354 28 silva silva PROPN ejde-39 354 29 department department PROPN ejde-39 354 30 of of ADP ejde-39 354 31 mathematics mathematics PROPN ejde-39 354 32 , , PUNCT ejde-39 354 33 federal federal ADJ ejde-39 354 34 university university NOUN ejde-39 354 35 of of ADP ejde-39 354 36 goiás goiás PROPN ejde-39 354 37 74001 74001 NUM ejde-39 354 38 - - SYM ejde-39 354 39 970 970 NUM ejde-39 354 40 , , PUNCT ejde-39 354 41 goiânia goiânia PROPN ejde-39 354 42 go go VERB ejde-39 354 43 , , PUNCT ejde-39 354 44 brazil brazil PROPN ejde-39 354 45 email email NOUN ejde-39 354 46 address address NOUN ejde-39 354 47 : : PUNCT ejde-39 354 48 eddomingos@hotmail.com eddomingos@hotmail.com X ejde-39 355 1 maxwell maxwell PROPN ejde-39 355 2 lizete lizete PROPN ejde-39 355 3 da da PROPN ejde-39 355 4 silva silva PROPN ejde-39 355 5 department department PROPN ejde-39 355 6 of of ADP ejde-39 355 7 mathematics mathematics PROPN ejde-39 355 8 , , PUNCT ejde-39 355 9 federal federal ADJ ejde-39 355 10 university university NOUN ejde-39 355 11 of of ADP ejde-39 355 12 goiás goiás PROPN ejde-39 355 13 74001 74001 NUM ejde-39 355 14 - - SYM ejde-39 355 15 970 970 NUM ejde-39 355 16 , , PUNCT ejde-39 355 17 goiânia goiânia PROPN ejde-39 355 18 go go VERB ejde-39 355 19 , , PUNCT ejde-39 355 20 brazil brazil PROPN ejde-39 355 21 email email NOUN ejde-39 355 22 address address NOUN ejde-39 355 23 : : PUNCT ejde-39 356 1 maxwelllizete@gmail.com maxwelllizete@gmail.com X ejde-39 356 2 1 1 X ejde-39 356 3 . . PUNCT ejde-39 356 4 introduction introduction NOUN ejde-39 356 5 2 2 NUM ejde-39 356 6 . . PUNCT ejde-39 356 7 preliminaries preliminary NOUN ejde-39 356 8 3 3 NUM ejde-39 356 9 . . PUNCT ejde-39 356 10 properties property NOUN ejde-39 356 11 of of ADP ejde-39 356 12 re re NOUN ejde-39 356 13 4 4 NUM ejde-39 356 14 . . PUNCT ejde-39 356 15 proof proof NOUN ejde-39 356 16 of of ADP ejde-39 356 17 theorem theorem NOUN ejde-39 356 18 ? ? PUNCT ejde-39 356 19 ? ? PUNCT ejde-39 357 1 5 5 X ejde-39 357 2 . . X ejde-39 357 3 conclusions conclusion NOUN ejde-39 357 4 and and CCONJ ejde-39 357 5 discussion discussion NOUN ejde-39 357 6 6 6 NUM ejde-39 357 7 . . PUNCT ejde-39 358 1 appendix appendix VERB ejde-39 358 2 acknowledgments acknowledgment NOUN ejde-39 358 3 references reference NOUN