Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 30, pp. 1–18. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu A WEIGHTED (p, 2)-EQUATION WITH DOUBLE RESONANCE ZHENHAI LIU, NIKOLAOS S. PAPAGEORGIOU Abstract. We consider a Dirichlet problem driven by a weighted (p, 2)-Laplacian with a reaction which is resonant both at ±∞ and at zero (double resonance). We prove a multiplicity theorem producing three nontrivial smooth solutions with sign information and ordered. In the appendix we develop the spectral properties of the weighted r-Laplace differential operator. 1. Introduction Let Ω ⊆ RN be a bounded domain with a C2-boundary ∂Ω. In this article we study the weighted (p, 2)-equation: −∆a1 p u(z)−∆a2u(z) = f(z, u(z)) in Ω, u|∂Ω = 0, 2 < p. (1.1) Given a ∈ L∞(Ω) with 0 < ĉ ≤ ess infΩ a and r ∈ (1,∞), we denote by ∆a r the weighted r-Laplace differential operator defined by ∆a ru = div(a(z)|Du|r−2Du), ∀u ∈W 1,r 0 (Ω). In problem (1.1) we have the sum of two such operators with different expo- nents. So,the differential operator driving the equation in (1.1) is not homogeneous and of course is space dependent. The reaction (right-hand side) of (1.1), is a Carathéodory function f(z, x) (that is, for all x ∈ R, z → f(z, x) is measurable and for a.a.z ∈ Ω, x → f(z, x) is continuous) which exhibits (p − 1) sublinear growth as x → ±∞ and resonance can occur with respect to the principal eigenvalue of (−∆a1 p ,W 1,p 0 (Ω)) (see the apendix). Also at zero, we can have resonance with respect to some nonprincipal eigenvalue of (−∆a2 , H1 0 (Ω)). So, our problem has double resonance. Using variational tools from the critical point theory together with truncation techniques and critical groups, we prove a multiplicity theorem for problem (1.1), producing three nontrivial smooth solutions, all with sign informa- tion and ordered. Recently a three solutions theorem for a superlinear weignted (p, q)-equation without resonance at zero, was proved by Liu-Papageorgiou [12], extending the well-known semilinear work of Wang [20]. Here we complement the aforementioned work of Liu-Papageorgiou [12], by examining the sublinear, double resonance case. 2020 Mathematics Subject Classification. 35J20, 35J60. Key words and phrases. Constant sign and nodal solutions; nonlinear regularity; nonlinear maximum principle; critical groups; spectrum of weighted r-Laplacian; double resonance. ©2023. This work is licensed under a CC BY 4.0 license. Submitted January 16, 2023. Published March 30, 2023. 1 2 Z. LIU, N. S. PAPAGEORGIOU EJDE-2023/30 Our hypotheses allow for resonance to occur as x → +∞ with respect to the principal eigenvalue of (−∆a1 p ,W 1,p 0 (Ω)) and as x → 0+ with respect to a higher eigenvalue of (−∆a2 , H1 0 (Ω)). So we have a double resonance situation which has not been examined in the past. 2. Mathematical background and hypotheses The analysis of problem (1.1) uses the Sobolev space W 1,p 0 (Ω) and the Banach space C1 0 (Ω) = {u ∈ C1(Ω) : u|∂Ω = 0}. The Poincaré inequality implies that on W 1,p 0 (Ω) we can use the equivalent norm ‖u‖ = ‖Du‖p for all u ∈W 1,p 0 (Ω). The Banach space C1 0 (Ω) is ordered with positive (order) cone C+ = {u ∈ C1 0 (Ω) : u(z) ≥ 0 for all z ∈ Ω}. This cone has a nonempty interior given by intC+ = {u ∈ C+ : u(z) > 0 for all z ∈ Ω, ∂u ∂n |∂Ω < 0} with ∂u ∂n = (Du,n)RN where n(·) is the outward unit normal on ∂Ω. If u : Ω → R is a measurable function, then we set u±(z) = max{±u(z), 0} for all z ∈ Ω. Both are measurable functions and we have u = u+ − u−, |u| = u+ + u− and if u ∈W 1,p 0 (Ω), then u± ∈W 1,p 0 (Ω). Let V : W 1,p 0 (Ω) → W−1,p′(Ω) ( 1 p + 1 p′ = 1) be the nonlinear operator defined by 〈V (u), h〉 = ∫ Ω [a1(z)|Du|p−2 + a2(z)|Du|q−2](Du,Dh)RN dz ∀u, h ∈W 1,p 0 (Ω). This operator is continuous and strictly monotone, thus maximal monotone too and of type (S)+ (see [6, p. 279]). Let X be a Banach space and ϕ ∈ C1(X,R). We say that ϕ(·) satisfies the “C-condition”, if the following property holds: Every sequence {un}n∈N ⊆ X such that {ϕ(un)}n∈N ⊆ R is bounded, and (1 + ‖un‖X)ϕ′(un) → 0 in X∗, admits a strongly convergent subsequence. A coercive functional ϕ ∈ C1(X,R) satisfies the C-condition (see Papageorgiou- Rădulescu-Repovš [16],p. 369). Given ϕ ∈ C1(X,R) and c ∈ R, we define the sets Kϕ = {u ∈ X : ϕ′(u) = 0}, ϕc = {u ∈ X : ϕ(u) ≤ c}. For a topological pair (Y2, Y1) with Y1 ⊆ Y2 ⊆ X and k ∈ N0, by Hk(Y2, Y1) we denote the kth-singular homology group with integer coefficients. Given u ∈ Kϕ isolated, the critical groups of ϕ at u, are defined by Ck(ϕ, u) = Hk(ϕc ∩ U,ϕc ∩ U \ {u}) for all k ∈ N0, with U a neighborhood of u such that Kϕ ∩ ϕc ∩ U = {u}.The excision property of singular homology, implies that the above definition of critical groups at u, is independent of the particular choice of the isolating neighborhood U . Suppose that ϕ ∈ C1(X,R) satisfies the C-condition and −∞ < inf ϕ(Kϕ). Then the critical groups of ϕ(·) at infinity are Ck(ϕ,∞) = Hk(X,ϕc) for all k ∈ N0, with c < inf ϕ(Kϕ). EJDE-2023/30 A WEIGHTED (p, 2)-EQUATION WITH DOUBLE RESONANCE 3 The Second Deformation Theorem (see [16, p. 386]) implies that the above definition is independent of the choice of the level c < inf ϕ(Kϕ). By λ̂a11 (p) we denote the first eigenvalue of (−∆a1 p ,W 1,p 0 (Ω)). We know that λ̂a11 (p) > 0 is simple, isolated and is the only eigenvalue of (−∆a1 p ,W 1,p 0 (Ω)) with eigenfunctions of constant sign. By û1(p) we denote the positive, Lp-normalized (that is, ‖û1(p)‖p = 1) eigenfunction corresponding to λ̂a11 (p). If a ∈ C0,1(Ω) (i.e. the space of all R-valued Lipschitz functions on Ω) and 0 < ĉ ≤ minΩa1, then the nonlinear regularity theory (see Lieberman [11]) and the nonlinear maximum principle (see Liu-Papageorgiou [13, 14]), imply that û1(p) ∈ intC+. We denote by {λ̂a2m }m∈N the sequence of distinct eigenvalues of (−∆a2 , H1 0 (Ω)). We know that λ̂a2n (2) → +∞ as n → ∞ and the sequence exhausts the set of eigenvalues of (−∆a2 , H1 0 (Ω)). In the appendix, we present in detail the main spectral properties of (−∆a1 p ,W 1,p 0 (Ω)) and of (−∆a2 , H1 0 (Ω)). The hypotheses on the data of (1.1) are as follows: (H0) Functions a1, a2 ∈ C0,1(Ω), and 0 < ĉ ≤ a1(z), a2(z) for all z ∈ Ω. (H1) f : Ω × R → R is a Carathéodory function such that f(z, 0) = 0 for a.a. z ∈ Ω and (i) |f(z, x)| ≤ a(z)[1+ |x|p−1] for a.a. z ∈ Ω, all x ∈ R, with a ∈ L∞(Ω)+; (ii) lim supx→±∞ f(z,x) |x|p−2x ≤ λ̂ a1 1 (p) uniformly for a.a. z ∈ Ω; (iii) if F (z, x) = ∫ x 0 f(z, s)ds, then there exists τ ∈ (2, p) such that lim x→±∞ f(z, x)x− pF (z, x) |x|τ = +∞ uniformly for a.a. z ∈ Ω; (iv) there exist m ∈ N,m ≥ 2, δ > 0 and η ∈ L∞(Ω) such that η(z) ≤ λ̂a2m+1(2) for a.a. z ∈ Ω, η 6≡ λ̂a2m+1(2), lim sup x→0 f(z, x) x ≤ η(z) uniformly for a.a. z ∈ Ω; λ̂a21 (2)x2 ≤ f(z, x)x for a.a. z ∈ Ω, all |x| ≤ δ. Remark 2.1. Hypothesis (H1)(ii) implies that we can have resonance with respect to λ̂a11 (p) as x → ±∞. Similarly, hypothesis H1(iv) allows for resonance to occur with respect to λ̂a2m (2)(m ≥ 2) as x→ 0. We introduce the energy functional for problem (1.1), ϕ : W 1,p 0 (Ω)→ R defined by ϕ(u) = 1 p ∫ Ω a1(z)|Du|p dz + 1 2 ∫ Ω a2(z)|Du|2 dz − ∫ Ω F (z, u) dz . Evidently ϕ ∈ C1(W 1,p 0 (Ω)). Also we introduce the positive and negative truncations of ϕ(·), namely the functionals ϕ± : W 1,p 0 (Ω)→ R defined by ϕ±(u) = 1 p ∫ Ω a1(z)|Du|p dz + 1 2 ∫ Ω a2(z)|Du|2 dz − ∫ Ω F (z,±u±) dz . Again we have ϕ± ∈ C1(W 1,p 0 (Ω)). 4 Z. LIU, N. S. PAPAGEORGIOU EJDE-2023/30 3. Three solution theorem In this section we prove that problem (1.1) has at least three nontrivial smooth solutions. Our approach uses variational and truncation techniques and critical groups. Proposition 3.1. Under hypotheses (H0), (H1), the functionals ϕ and ϕ± are coercive. Proof. We do the proof for ϕ+(·), the proofs for ϕ(·), ϕ−(·) being similar. We argue indirectly. So, suppose that ϕ+(·) is not coercive. We can find {un}n∈N ⊆W 1,p 0 (Ω) such that ϕ+(un) ≤ c1 for some c1 > 0 and all n ∈ N, ‖un‖ → ∞. (3.1) If {u+ n }n∈N ⊆W 1,p 0 (Ω) is bounded, then from the inequality in (3.1) and hypothesis (H1)(i) we have {u−n }n∈N ⊆ W 1,p 0 (Ω) is bounded; therefore {un}n∈N ⊆ W 1,p 0 (Ω) is bounded, a contradiction to (3.1). So, we can say that ‖u+ n ‖ → ∞. (3.2) Let yn = u+ n ‖u+ n ‖ , n ∈ N. We have ‖yn‖ = 1, yn ≥ 0 for all n ∈ N. We can assume that yn w→ y in W 1,p 0 (Ω), yn → y in Lp(Ω), y ≥ 0. (3.3) From the inequality in (3.1), we have 1 p ∫ Ω a1(z)|Dyn|p dz + 1 2‖u+ n ‖p−2 ∫ Ω a2(z)|Dyn|2 dz ≤ c1 + ∫ Ω F (z, u+ n ) ‖u+ n ‖p dz (3.4) for all n ∈ N. Using hypothesis (H1)(i), we have |F (z, u+ n (z))| ‖u+ n ‖p ≤ c2[1 + yn(z)p] for a.a. z ∈ Ω and all n ∈ N, some c2 > 0, which implies {F (·, u+ n (·)) ‖u+ n ‖p } n∈N ⊆ L 1(Ω) is uniformly integrable. Then invoking the Dunford-Pettis theorem (see Papageorgiou-Winkert [18, p. 289]), we can say that at least for a subsequence, F (·, u+ n (·)) ‖u+ n ‖p w→ 1 p η̂ in L1(Ω). (3.5) Hypothesis (H1)(ii) implies that η̂(z) = ϑ(z)y(z)p for a.a. z ∈ Ω, (3.6) with ϑ ∈ L∞(Ω), ϑ(z) ≤ λ̂a11 (p) for a.a. z ∈ Ω (see Aizicovici-Papageorgiou-Staicu [1], proof of Proposition 16). If in (3.4) we pass to the limit as n → ∞ and use (3.2) (recall 2 < p), (3.3), (3.5), (3.6), we obtain∫ Ω a1(z)|Dy|p dz ≤ ∫ Ω η(z)yp dz. (3.7) First assume that ϑ 6≡ λ̂a11 (p). Using Proposition 4.1 in the appendix, we have c3‖y‖p ≤ ∫ Ω a1(z)|Dy|p dz − ∫ Ω ϑ(z)yp dz for some c3 > 0. (3.8) EJDE-2023/30 A WEIGHTED (p, 2)-EQUATION WITH DOUBLE RESONANCE 5 From (3.7) and (3.8) it follows that y = 0. But then from (3.4) we have∫ Ω a1(z)|Dyn|p dz → 0 which implies yn → 0 in W 1,p 0 (Ω) (see hypotheses (H0)). This contradicts that ‖yn‖ = 1 for all n ∈ N. Next we assume that ϑ(z) = λ̂a11 (p) for a.a. z ∈ Ω. From (3.7) and the variational characterization of λ̂a11 (p) > 0 (see (4.2) in the appendix), we have∫ Ω a1(z)|Dy|p dz = λ̂a11 (p)‖y‖pp, which implies y = βû1(p) with β ≥ 0 (recall that y ≥ 0). If β = 0, then y = 0 and as above, we have yn → 0 in W 1,p 0 (Ω), which contradic- tion that ‖yn‖ = 1 for all n ∈ N. Hence y = βû1(p) with β > 0 and so y ∈ intC+ (since û1(p) ∈ intC+, see hypotheses (H0)). Therefore, u+ n (z)→ +∞ for a.a. z ∈ Ω. (3.9) Hypothesis (H1)(iii) implies that given any M > 0, we can find γ > 0 such that Mxτ ≤ f(z, x)x− pF (z, x) for a.a. z ∈ Ω, and all x ≥ γ. (3.10) We have d dx ( F (z, x) xp ) = f(z, x)xp − pxp−1F (z, x) x2p = f(z, x)x− pF (z, x) xp+1 ≥ M xp+1−τ for a.a. z ∈ Ω, and all x ≥ γ (see (3.10)); therefore, F (z, v) vp − F (z, x) xp ≥ − M p− τ [ 1 vp−τ − 1 xp−τ ] for a.a. z ∈ Ω and all v ≥ x ≥ γ > 0. Letting v → +∞ and using (H1)(ii), We obtain 1 p λ̂a11 (p)− F (z, x) xp ≥ M p− τ 1 xp−τ for a.a. z ∈ Ω and all x ≥ γ, ⇒ λ̂a11 (p)xp − pF (z, x) ≥ M p− τ xτ for a.a. z ∈ Ω and all x ≥ γ, ⇒ λ̂a11 (p)xp − pF (z, x) xτ ≥ M p− τ for a.a. z ∈ Ω and all x ≥ γ. Since M > 0 is arbitrary, it follows that λ̂a11 (p)xp − pF (z, x) xτ → +∞ as x→∞, uniformly for a.a. z ∈ Ω. (3.11) From (3.1) and (4.2) (in the appendix), we have∫ Ω [λ̂a11 (p)(u+ n )p − pF (z, u+ n )] dz ≤ pc1 for all n ∈ N, ⇒ ∫ Ω λ̂a11 (p)(u+ n )p − pF (z, u+ n ) (u+ n )τ yτn dz ≤ pc1 ‖u+ n ‖τ for all n ∈ N. (3.12) 6 Z. LIU, N. S. PAPAGEORGIOU EJDE-2023/30 Passing to the limit as n → ∞ in (3.12) and using (3.2), (3.9), (3.11) and Fatou’s lemma, we reach a contradiction. This proves that {u+ n }n∈N ⊆W 1,p 0 (Ω) is bounded. From the first part of the proof this implies that {un}n∈N ⊆ W 1,p 0 (Ω) ia bounded, contradicting (3.1). Therefore ϕ+(·) is coercive. Similarly for ϕ(·) and ϕ−(·). � Using Proposition 3.1, we can produce two constant sign smooth solutions. Proposition 3.2. Under hypotheses (H0), (H1), problem (1.1) has two constant sign solutions u0 ∈ intC+ and v0 ∈ − intC+, which are local minimizers of ϕ(·). Proof. From Proposition 3.1 we know that ϕ+(·) is coercive. Also, using the Sobolev embedding theorem, we see that ϕ+(·) is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find u0 ∈W 1,p 0 (Ω) such that ϕ+(u0) = inf[ϕ+(u) : u ∈W 1,p 0 (Ω)]. (3.13) Recall that û1(2) ∈ intC+ (see the appendix). Therefore, we can find t ∈ (0, 1) small such that 0 ≤ tû1(2)(z) ≤ δ for all z ∈ Ω, (3.14) where δ > 0 is as postulated by hypothesis H1(iv). Then, using (3.14) and hypoth- esis (H1)(iv), we have ϕ+(tû1(2)) ≤ tp p ∫ Ω a1(z)|Dû1(2)|p dz + t2 2 [λ̂a21 (2)− λ̂a2m (2)] = c4t p − c5t2 for some positive constants c4 and c5 > 0. Here we used that ‖û1(2)‖2 = 1 and that m ≥ 2. Since 2 < p, choosing t ∈ (0, 1) and small, we have ϕ+(tû1(2)) < 0, which implies ϕ+(u0) < 0 = ϕ+(0) (see (3.13)); thus u0 6= 0. From (3.13) we have ϕ′+(u0), h〉 = 0 for all h ∈W 1,p 0 (Ω), which implies 〈V (u0), h〉 = ∫ Ω f(z, u+ 0 )h dz for all h ∈W 1,p 0 (Ω). (3.15) In (3.15) we choose h = −u−0 ∈W 1,p 0 (Ω) and obtain ĉ‖Du−0 ‖pp ≤ 0 (see hypotheses (H0) ⇒ u0 ≥ 0, u0 6= 0. Then from (3.15) we have −∆a1 p u0 −∆a2u0 = f(z, u0) in Ω. (3.16) From Ladyzhenskaya-Uraltseva [10, p. 286], we have u0 ∈ L∞(Ω). Then the non- linear regularity theory of Lieberman [11] implies that u0 ∈ C+ \ {0}. On account of hypotheses (H1)(i),(iv), we can find c6 > 0 such that f(z, x) ≥ λ̂a2m (2)x− c6xp−1 for a.a. z ∈ Ω and all x ≥ 0. So, if ϑ̂ > c6, then f(z, x) + ϑ̂xp−1 ≥ 0 for a.a. z ∈ Ω and all z ≥ 0. From (3.16) we have −∆a1 p u0 −∆a2u0 + ϑ̂up−1 0 ≥ 0 in Ω, which implies u0 ∈ intC+, see [13, Lemma 1]. Note that ϕ|C+ = ϕ+|C+ . EJDE-2023/30 A WEIGHTED (p, 2)-EQUATION WITH DOUBLE RESONANCE 7 So u0 is a local C1 0 (Ω) minimizer of ϕ(·) and from [17, Proposition A3], we conclude that u0 is a local W 1,p 0 (Ω)-minimizer of ϕ(·). Similarly, working now with ϕ−(·), we produce a negative solution v0 ∈ − intC+, which is a local minimizer of ϕ(·). � We assume that Kϕ is finite or otherwise we already have an infinity of solutions of (1.1) and so we are done. Then from Proposition 3.2 and [16, Proposition 6.2.3], we have the following result. Corollary 3.3. If (H0), (H1) hold and u0 ∈ intC+ and v0 ∈ − intC+ are the two constants sign solutions from Proposition 3.2, then Ck(ϕ, u0) = Ck(ϕ, v0) = δk,0Z for all k ∈ N0. In what follows we denote by E(λ̂a21 (2)) the eigenspace corresponding is the eigen- value λ̂a2i (2), i ∈ N. We know that E(λ̂a2i (2)) is finite dimensional and E(λ̂a2i (2)) ⊆ C1 0 (Ω) (see the appendix). Proposition 3.4. If hypotheses (H0), (H1) hold, then Cdm(ϕ, 0) 6= 0, where dm = dimHm with Hm = ⊕mi=1E(λ̂a21 (2)). Proof. We consider the C1-functional ψ : H1 0 (Ω)→ R defined by ψ(u) = 1 2 ∫ Ω a2(z)|Du|2 dz − ∫ Ω F (z, u) dz . Let u ∈ Hm. Since Hm ⊆ C(Ω) is finite dimensional, all norms are equivalent and so we can find ρ > 0 such that u ∈ Hm and ‖u‖ ≤ ρ ⇒ |u(z)| ≤ δ for all z ∈ Ω, with δ > 0 as postulated by hypothesis (H1)(iv). So for u ∈ Hm with ‖u‖ ≤ ρ, we have ψ(u) ≤ 1 2 [‖Du‖22 − λ̂a2m (2)‖u‖22] ≤ 0 (3.17) (see hypothesis (H1)(iv) and the appendix). On account of hypotheses (H1)(i),(iv), given ε > 0, we can find cε > 0 such that F (z, x) ≤ 1 2 [η(z) + ε]x2 + cε|x|p for a.a. z ∈ Ω and all x ∈ R. (3.18) Then for u ∈ Ĥm+1 = H ⊥ m, from (3.18), we have ψ(u) ≥ 1 2 [ ∫ Ω a2(z)|Du|2 dz − ∫ Ω η(z)u2 dz − ε‖u‖22]− ĉε‖u‖p for some ĉε > 0 ≥ 1 2 [c7 − ε λ̂a2m+1 ]‖u‖2 − ĉε‖u‖p for some positive constant c7 (see Proposition 4.2). Choosing ε ∈ (0, λ̂a2m+1(2)c7), we obtain ψ(u) ≥ c8‖u‖2 − ĉε‖u‖p for some c8 > 0. So, we can find ρ0 ∈ (0, ρ] such that ψ(u) > 0 for all u ∈ Ĥm+1, 0 < ‖u‖ ≤ ρ0. (3.19) 8 Z. LIU, N. S. PAPAGEORGIOU EJDE-2023/30 Then (3.17) and (3.19) imply that ψ(·) has a local linking at u = 0. So, invoking [16, Theorem 6.6.17], we have Cdm(ψ, 0) 6= 0. Let ψ̂ = ψ|W 1,p 0 (Ω). Since W 1,p 0 (Ω) ↪→ H1 0 (Ω) continuously and densely, using [15, Theorems 16 and 17] (see also Chang [12, p. 14]), we have Ck(ψ̂, 0) = Ck(ψ, 0) for all k ∈ N0, (3.20) which implies Cdm(ψ̂, 0) 6= 0. Note that |ϕ(u)− ψ̂(u)| = 1 p ∫ Ω a1(z)|Du|p dz ≤ ‖a1‖∞ p ‖u‖p. (3.21) Also for all h ∈W 1,p 0 (Ω), we have |〈ϕ′(u)− ψ̂′(u), h〉| = ∫ Ω a1(z)|Du|p−2(Du,Dh)RN dz ≤ ‖a1‖∞ ∫ Ω |Du|p−1|Dh| dz ≤ ‖a1‖∞‖Du‖p−1 p ‖Dh‖p; therefore, ‖ϕ′(u)− ψ̂′(u)‖ ≤ c9‖u‖p−1 for some c9 > 0. (3.22) Recall that we assume Kϕ is finite (otherwise we already have an infinity of distinct nontrivial smooth positive solutions and so we are done). Then from (3.21), (3.22) and the C1-continuity property of critical groups (see Gasiński-Papageorgiou [6, p. 836]), we have Ck(ϕ, 0) = Ck(ψ̂, 0) for all k ∈ N0, which implies Cdm(ϕ, 0) 6= 0 (see (3.20)). � Next we will produce a third nontrivial solution which is nodal. To do this, we need some auxiliary results. Note that hypotheses (H1)(i),(iv) imply that we can find c10 > 0 such that f(z, x)x ≥ λ̂a2mx2 − c10|x|p for a.a. z ∈ Ω and all x ∈ R. (3.23) Then (3.23) leads to the auxiliary Dirichlet problem −∆a1 p u(z)−∆a2u(z) = λ̂a2m (2)u(z)− c10|u(z)|p−2u(z) in Ω, u|∂Ω = 0. (3.24) Reasoning as in [13, Proposition 3], we have the following result. Proposition 3.5. If hypotheses (H0) holds and m ≥ 2, then problem (3.24) has a unique positive solution u ∈ intC+ and since the equation is odd v = −u ∈ − intC+ is the unique negative solution of problem (3.24). Using Proposition 3.5, we can produce extremal constant sign solutions for prob- lem (1.1), that is, a smallest positive solution and a biggest negative solution. EJDE-2023/30 A WEIGHTED (p, 2)-EQUATION WITH DOUBLE RESONANCE 9 Proposition 3.6. If hypotheses (H0), (H1) hold, then problem (1.1) has a smallest positive solution û ∈ intC+ and a biggest negative solution v̂ ∈ − intC+. Proof. Let S+ (resp. S−) denote the set of positive (resp. negative) solutions of (1.1). From Proposition 3.2 and its proof, we know that ∅ 6= S+ ⊆ intC+ and ∅ 6= S− ⊆ − intC+. Moreover, we know (see Filippakis-Papageorgiou [4]) that S+ is downward directed (that is, if u1, u2 ∈ S+, then there exists u ∈ S+ such that u ≤ u1, u ≤ u2), S− is upward directed (that is, if v1, v2 ∈ S−, then there exists v ∈ S− such that v1 ≤ v, v2 ≤ v). Next we show that u ≤ u for all u ∈ S+, v ≤ v for all v ∈ S−. (3.25) To this end, let u ∈ S+ ⊆ intC+ and introduce the Caratheodory function k+ : Ω× R→ R defined k+(z, x) = { λ̂a2m (2)x+ − c10(x+)p−1 if x ≤ u(z) λ̂a2m (2)u(z)− c10(u(z))p−1 if u(z) ≤ x. (3.26) We set K+(z, x) = ∫ x 0 k+(z, s)ds and consider the C1-functional σ+ : W 1,p 0 (Ω)→ R defined by σ+(u) = 1 p ∫ Ω a1(z)|Du|p dz + 1 2 ∫ Ω a2(z)|Du|2 dz − ∫ Ω K+(z, u) dz. From hypotheses H0 and (3.26), we see that σ+(·) is coercive. Also, by the Sobolev embedding theorem σ+(·) is sequentially weakly lower semicontinuous. So, we can find ũ ∈W 1,p 0 (Ω) such that σ+(ũ) = inf[σ+(u) : u ∈W 1,p 0 (Ω)]. (3.27) Since u ∈ intC+, we can find t ∈ (0, 1) small such that tû1(2) ≤ u (see [16, Proposition 4.1.22]. Then using (3.26) and since m ≥ 2 we have (by taking t ∈ (0, 1) and small, σ+(tû1(2)) < 0 which implies σ+(ũ) < 0 = σ+(0) (see (3.27)); thus, ũ 6= 0. From (3.27) we have that 〈σ′+, h〉 = 0 for all h ∈W 1,p 0 (Ω). Therefore, 〈V (ũ), h〉 = ∫ Ω k+(z, ũ)h dz for all h ∈W 1,p 0 (Ω). (3.28) In (3.28) first we use the test function h = −ũ− ∈W 1,p 0 (Ω) and obtain ĉ‖Dũ−‖pp ≤ 0, hence ũ ≥ 0, ũ 6= 0. Next choosing h = [ũ− u]+ ∈W 1,p 0 (Ω) in (3.28), we have 〈V (ũ), (ũ− u)+〉 = ∫ Ω [λ̂a2m (2)u− c10u p−1](ũ− u)+ dz (see (3.26)) 10 Z. LIU, N. S. PAPAGEORGIOU EJDE-2023/30 ≤ ∫ Ω f(z, u)(ũ− u)+ dz (see (3.23)) = 〈V (u), (ũ− u)+〉 (since u ∈ S+), which implies ũ ≤ u, from the monotonicity of V (·)). So, we have proved that ũ ∈ [0, u], ũ 6= 0. (3.29) Then (3.29), (3.26), (3.28), and Proposition 3.5 imply that ũ = u ∈ intC+. This in turn implies u ≤ u for all u ∈ S+. Similarly, we can show that v ≤ v for all v ∈ S−. Using [8, Theorem 5.109], we can find {un}n∈N ⊆ S+ decreasing (since S+ is downward directed) such that inf S+ = inf n∈N un. We have 〈V (un), h〉 = ∫ Ω f(z, un)h dz for all h ∈W 1,p 0 (Ω) and all n ∈ N, (3.30) u ≤ un ≤ u1 for all n ∈ N (see (3.25)). (3.31) Choosing h = un ∈ W 1,p 0 (Ω) in (3.30) and using (3.31) and (H1)(i), we infer that {un}n∈N ⊆W 1,p 0 (Ω) is bounded. So, we may assume that un w→ û in W 1,p 0 (Ω), un → û in Lp(Ω), as n→∞. (3.32) In (3.30) we use h = un− û ∈W 1,p 0 (Ω), pass to the limit as n→∞ and use (3.32). We obtain limn→∞〈V (un), un − û〉 = 0 which implies un → û in W 1,p 0 (Ω) (the (S)+-property of V (·)). (3.33) Passing to the limit as n→∞ in (3.30) and using (3.33), we obtain 〈V (û), h〉 = ∫ Ω f(z, û)h dz for all h ∈W 1,p 0 (Ω), u ≤ û (see (3.31)), Therefore, û ∈ S+ ⊆ intC+ and û = inf S+. Similarly for S− which is upward directed and so the sequence {vn}n∈N ⊆ S− such that supS− = supn∈N vn, will be increasing. � Now we try to produce a nontrivial solution of (1.1) in the order interval [v̂, û] = {h ∈W 1,p 0 (Ω) : v̂(z) ≤ h(z) ≤ û(z) for a.a. z ∈ Ω}. On account of the extremality of û and v̂ any such solution distinct from û and v̂ will be nodal. To this end, we introduce the following truncation of the reaction f(z, ·) e(z, x) =  f(z, v̂(z)) if x < v̂(z) f(z, x) if v̂(z) ≤ x ≤ û(z) f(z, û(z)) if û(z) < x. (3.34) Also, we consider the positive and negative truncations of e(z, ·), namely the func- tions e±(z, x) = e(z,±x±). (3.35) EJDE-2023/30 A WEIGHTED (p, 2)-EQUATION WITH DOUBLE RESONANCE 11 All three functions are Caratheodory. We set E(z, x) = ∫ x 0 e(z, s)ds, E±(z, x) = ∫ x 0 e±(z, s)ds. Then we consider the C1-functionals w,w± : W 1,p 0 (Ω)→ R defined by w(u) = 1 p ∫ Ω a1(z)|Du|p dz + 1 2 ∫ Ω a2(z)|Du|2 dz − ∫ Ω E(z, u) dz, w±(u) = 1 p ∫ Ω a1(z)|Du|p dz + 1 2 ∫ Ω a2(z)|Du|2 dz − ∫ Ω E±(z, u) dz . Using (3.34) and (3.35), we can easily show that Kw ⊆ [v̂, û] ∩ C1 0 (Ω),Kw+ ⊆ [0, û] ∩ C+,Kw− ⊆ [v̂, 0] ∩ (−C+). The extremality of û, v̂ implies that Kw ⊆ [v̂, û] ∩ C1 0 (Ω), Kw+ = {0, û},Kw− ⊆ {v̂, 0}. (3.36) Now, we can generate the third nontrivial smooth solution of (1.1) which is nodal. By intC1 0 (Ω)[v̂, û] we denote the interior in C1 0 (Ω) of [v̂, û] ∩ C1 0 (Ω). Proposition 3.7. If (H0), (H1) hold, then problem (1.1) has a nodal solution y0 ∈ [v̂, û] ∩ C1 0 (Ω). Proof. First we show that û, v̂ are local minimizers of w(·). To this end, note that w+(·) is coercive (see (H0) and (3.34), (3.35)). Also, it is sequentially weakly lower semicontinuous. So, we can find ũ ∈W 1,p 0 (Ω) such that w+(ũ) = inf[w+(u) : u ∈W 1,p 0 (Ω)]. (3.37) Since û ∈ intC+ as before for t ∈ (0, 1) small (at least such that tû1(2) ≤ û), we have w+(±û1(2)) < 0, ⇒ w+(ũ) < 0 = w+(0), ⇒ ũ 6= 0. (3.38) From (3.37),(3.38) and (3.36) we infer that ũ = û ∈ intC+. Note that w|C+ = w+|C+ , ⇒ û is a local C1 0 (Ω)-minimizer of w(·) ⇒ û is a local W 1,p 0 (Ω)-minimizer of w(·) ; see Papageorgiou-Rădulescu-Zhang [17, Proposition A3]. Similarly for v̂ ∈ − intC+ using the functional w−(·), we have Ck(w, û) = Ck(w, v̂) = δk,0Z for all k ∈ N0. (3.39) Next we show that Cdm(w, 0) 6= 0. (3.40) Consider the homotopy h(t, u) = (1− t)ϕ(u) + tw(u) for all (t, u) ∈ [0, 1]×W 1,p 0 (Ω). Suppose we can find {(tn, un)}n∈N ⊆ [0, 1]×W 1,p 0 (Ω) such that tn → t in [0, 1], un → 0 in W 1,p 0 (Ω), h′u(tn, un) = 0 for all n ∈ N. (3.41) 12 Z. LIU, N. S. PAPAGEORGIOU EJDE-2023/30 From the equation in (3.41) we have 〈V (un), h〉 = ∫ Ω [(1− tn)f(z, un) + tne(z, un)]h dz (3.42) for all h ∈W 1,p 0 (Ω) and all n ∈ N, From (3.41), (3.42), and [10, Theorem 7.1], we know that we can find c11 > 0 such that un ∈ L∞(Ω), ‖un‖∞ ≤ c11 for all n ∈ N. Then the nonlinear regularity by Lieberman [11], implies that there exist α ∈ (0, 1) and c12 > 0 such that un ∈ C1,α 0 (Ω), ‖un‖C1,α 0 (Ω) ≤ c12, for all n ∈ N. (3.43) Since C1,α 0 (Ω) ↪→ C1 0 (Ω) compactly, from (3.43) and (3.41) it follows that un → u in C1 0 (Ω). This implies un ∈ intC1 0 (Ω)[v̂, û] for all n ≥ n0 (recall û ∈ intC+ and v̂ ∈ − intC+). Therefore, {un}n≥n0 ⊆ Kϕ; see (3.34)). But we have assumed that Kϕ is finite (see the proof of Proposition 3.4. There- fore (3.41) can not be true and then the homotopy invariance property of critical groups (see [6, p. 836]) implies Ck(ϕ, 0) = Ck(w, 0) for all k ∈ N0, (3.44) Then (3.44) and Proposition 3.4 imply that (3.40) is true. Evidently w(·) is coercive. Hence Ck(w,∞) = δk,0Z for all k ∈ N0; (3.45) see [16, Proposition 6.2.24]. From (3.40), (3.45), and [16, Corollary 6.7.8], we know that there exists y0 ∈ Kw ⊆ [v̂, û] ∩ C1 0 (Ω) such that w(y0) < 0 = w(0) and Cdm−1(w, y0) 6= 0, or w(y0) > 0 = w(0) and Cdm+1(w, y0) 6= 0. (3.46) Evidently y0 6= 0. Since m ≥ 2, we have that dm ≥ 2. Therefore comparing (3.39) and (3.46), we conclude that y0 /∈ {û, v̂}. So, finally we have y0 ∈ [v̂, û] ∩ C1 0 (Ω) and y0 /∈ {0, û, v̂} which imply that y0 is a nodal solution of (1.1). � If we impose an additional condition on f(z, ·), we can improve the conclusion of the previous proposition. The new hypotheses on the reaction f(z, x) are as follows: (H2) For every ρ > 0, there exists θ̂ρ > 0 such that for a.a. z ∈ Ω, the mapping x 7→ f(z, x) + θ̂ρx p−1 is nondecreasing on [−ρ, ρ]. Proposition 3.8. If hypotheses (H0)–(H2) hold, then problem (1.1) has a nodal solution y0 ∈ intC1 0 (Ω)[v̂, û]. Proof. From Proposition 3.7, we already have a nodal solution y0 such that y0 ∈ [v̂, û] ∩ C1 0 (Ω). (3.47) Let γ(z, y) = a1(z)|y|p−2y + a2(z)y for all z ∈ Ω, y ∈ RN . For every u ∈ W 1,p 0 (Ω) we have −div γ(z,Du) = −∆a1 p u−∆a2u. EJDE-2023/30 A WEIGHTED (p, 2)-EQUATION WITH DOUBLE RESONANCE 13 Note that ∇yγ(z, y) = a1(z)|y|p−2[id+ (p− 2) y ⊗ y |y|2 ] + a2(z)id ∀z ∈ Ω, ;∀y ∈ RN ⇒ (∇yγ(z, y)β, β)RN ≥ ĉ|β|2 for all y, β ∈ RN . Then the tangency principle of Pucci-Serrin [19, p. 35] implies that y0(z) < û(z) for all z ∈ Ω. (3.48) Let ρ = max { ‖û‖∞, ‖v̂‖∞ } and let θ̂ρ > 0 be as postulated by hypothesis (H1) and (H2). Choose θ∗ρ > θ̂ρ, we have −∆a1 p û−∆a2 û+ θ∗ρû p−1 = f(z, û) + θ∗ρû p−1 = f(z, û) + θ̂ρû p−1 + (θ∗ρ − θ̂ρ)ûp−1 ≥ f(z, y0) + θ̂ρy p−1 0 + (θ∗ρ − θρ)y p−1 0 = f(z, y0) + θ∗ρy p−1 0 = −∆a1 p y0 −∆a2y0 + θ∗ρy p−1 0 , (3.49) where we used (3.47), (H1), and (H2). Since û ∈ intC+, y0 ∈ C1 0 (Ω), from (3.48), we see that for every K ⊆ Ω compact, we have 0 < cK ≤ û(z)− y0(z) for all z ∈ K. (3.50) Then (3.49), (3.50) and [7, Proposition 3.2] imply that û − y0 ∈ intC+. Similarly we show that y0 − v̂ ∈ intC+. Therefore finally we have y0 ∈ intC1 0 (Ω)[v̂, û]. � Concluding we can state the following multiplicity theorem for problem (1.1). We emphasize that we provide sign information for all the solutions and the three solutions are ordered. Theorem 3.9. (a) If (H0), (H1) hold, then problem (1.1) has at least three solutions u0 ∈ intC+, v0 ∈ − intC+, y0 ∈ [v0, u0] ∩ C1 0 (Ω) nodal. (b) If hypotheses (H0)–(H2) hold, then problem (1.1) has at least three solutions u0 ∈ intC+, v0 ∈ − intC+, y0 ∈ intC1 0 (Ω)[v0, u0] nodal. 4. Appendix In this section we present some basic facts concerning the spectral properties of (−∆a r ,W 1,r 0 (Ω)) and a ∈ C0,1(Ω), a(z) ≥ ĉ > 0 for all z ∈ Ω, 1 < r < ∞. We consider the nonlinear eigenvalue problem −∆a ru(z) = λ̂|u(z)|r−2u(z) in Ω, u|∂Ω = 0. (4.1) We say that λ̂ ∈ R is an eigenvalue of (−∆a r ,W 1,r 0 (Ω)) if problem (4.1) has a nontrivial weak solution û ∈ W 1,p 0 (Ω) known as an eigenfunction corresponding to the eigenvalue λ̂. Evidently every eigenvalue λ̂ ≥ 0. We show that there is a smallest eigenvalue λ̂a1(r) > 0. To see this, we minimize the Rayleigh quotient R(u) = ∫ Ω a(z)|Du|r dz ‖u‖rr , u ∈W 1,r 0 (Ω), u 6= 0. 14 Z. LIU, N. S. PAPAGEORGIOU EJDE-2023/30 We have 0 ≤ λ̂a1(r) = inf[ ∫ Ω a(z)|Du|r dz ‖u‖rr , u ∈W 1,r 0 (Ω), u 6= 0] = inf [ ∫ Ω a(z)|Du|r dz, u ∈W 1,r 0 (Ω), ‖u‖r = 1 ] , (4.2) by homogeneity. The infimum in (4.2) is attained. Consider a sequence {un}n∈N ⊆ W 1,r 0 (Ω) such that∫ Ω a(z)|Dun|r dz ↓ λ̂a1(r), ‖un‖r = 1 for all n ∈ N. Therefore {un}n∈N ⊆W 1,p 0 (Ω) is bounded. So, we can assume that un w→ û1 in W 1,r 0 (Ω), un → û1 in Lr(Ω). We have that∫ Ω a(z)|Dû1|r dz ≤ lim inf n→∞ ∫ Ω a(z)|Dun|r dz = λ̂a1(r), ‖û1‖r = 1, implies ∫ Ω a(z)|Dû1|r dz = λ̂a1(r) > 0, ‖û1‖r = 1. From (4.2) and the Lagrange multiplier rule, we infer that λ̂a1(r) > 0 is the small- est eigenvalue of (−∆a r ,W 1,r 0 (Ω)). Evidently we can replace û1 ∈ W 1,p 0 (Ω) by |û1| ∈ W 1,p 0 (Ω). Therefore we can always assume that û1 ≥ 0. The nonlinear regularity theory (see [11]) and the nonlinear maximum principle [13, 19], imply that û1 ∈ intC+. In fact λ̂a1(r) > 0 is the only eigenvalue with eigenfunctions of constant sign. All other eigenvalues have eigenfunctions which are nodal (sign- changing). The proof of this fact is done along the lines of the corresponding result for (−∆r,W 1,r 0 (Ω)) (see for example Gasiński-Papageorgiou [5, p. 743]). Suppose û and v̂ are two eigenfunctions corresponding of λ̂a1(r) > 0. As above, we have that û, v̂ ∈ intC+. Then using the nonlinear Picone’s identity of Jaros [9], we have 0 ≤ a(z)|Dû|r − a(z)|Dv̂|r−2(Dv̂,D( ûr v̂r−1 ))RN for all z ∈ Ω ⇒ 0 ≤ ∫ Ω a(z)|Dû|r dz − ∫ Ω −(∆a r v̂) ûr v̂r−1 dz (using Green’s identity, see [16, p. 34]) = ∫ Ω a(z)|Dû|r dz − λ̂a1(r)‖û‖rr = 0, which implies ûDv̂ = v̂Dû (see Jaros [9]). In turn this implies D( ûv̂ ) = 0 and so û = ϑv̂ with ϑ > 0. Then λ̂a1(r) > 0 is simple. Also λ̂a1(r) > 0 is isolated in the spectrum of (−∆a r ,W 1,r 0 (Ω)). Indeed, if λ̂a1(r) is not isolated, we can find λ̂n ↓ λ̂a1(r) with λ̂n eigenvalue of (−∆a r ,W 1,r 0 (Ω)) for every n ∈ N. Let ûn ∈ W 1,p 0 (Ω) be an eigenfunction corresponding to λ̂n. We have −∆a r ûn = λ̂n|ûn|r−2ûn for all n ∈ N. (4.3) EJDE-2023/30 A WEIGHTED (p, 2)-EQUATION WITH DOUBLE RESONANCE 15 Normalizing we may assume that ‖ûn‖r = 1 for all n ∈ N. Then from (4.3) it follows that {un}n∈N ⊆W 1,r 0 (Ω) is bounded. We can assume that ûn w→ û∗ in W 1,r 0 (Ω), ûn → û∗ in Lr(Ω). (4.4) We have ‖û∗‖r = 1. On (4.3) we act with ûn − û∗ ∈ W 1,r 0 (Ω), pass to the limit as n→∞ and use (4.4). We obtain lim n→∞ 〈Aar(ûn), ûn − û∗〉 = 0 (4.5) with Aar : W 1,r 0 (Ω) → W−1,r′(Ω) = W 1,r 0 (Ω)∗( 1 r + 1 r′ = 1) being the nonlinear operator defined by 〈Aar(u), h〉 = ∫ Ω a(z)|Du|r−2(Du,Dh)RN dz for all u, h ∈W 1,r 0 (Ω). This operator has the same properties as V (·). In particular, (4.5) and the (S)+- property of Aar(·), imply that ûn → û∗ in W 1,r 0 (Ω). (4.6) If in (4.3) we pass to the limit as n→∞, we have −∆a r û∗ = λ̂a1(r)|û∗|r−2û∗ in Ω, ‖û∗‖r = 1. Then we can assume that û∗ ∈ intC+. From (4.6) and the nonlinear regularity theory of Lieberman [11], we know that there exist α ∈ (0, 1) and M > 0 such that ûn ∈ C1,α 0 (Ω) and ‖ûn‖C1,α 0 (Ω) ≤M for all n ∈ N. (4.7) Then the compact embedding of C1,α 0 (Ω) into C1 0 (Ω) and (4.6) imply that ûn → û∗ in C1 0 (Ω). Since û∗ ∈ intC+, we have {un}n≥n0 ⊆ C+ \ {0}, which contradicts that only λ̂a1(r) > 0 has eigenfunctions of constant sign. Summarizing, we can state the following basic facts about the spectrum of (−∆a r ,W 1,r 0 (Ω)): • There is a smallest eigenvalue λ̂a1(r) > 0 which has a variational character- ization given by (4.2). • λ̂a1(r) is simple, isolated and the corresponding eigenfunctions are of con- stant sign and belong in (intC+) ∪ (− intC+). • If λ̂ > λ̂a1(r) is an eigenvalue, then λ̂ has nodal eigenfunctions. Proposition 4.1. If η ∈ L∞(Ω) and η(z) ≤ λ̂a1(r) for a.a. z ∈ Ω, η 6≡ λ̂a1(r), then there exists θ > 0 such that θ‖u‖r ≤ ∫ Ω a(z)|Du|r dz − ∫ Ω η(z)|u|r dz for all u ∈W 1,r 0 (Ω). Proof. Arguing by contradiction, suppose we can find {un}n∈N ⊆ W 1,p 0 (Ω) such that for all n ∈ N, we have∫ Ω a(z)|Dun|r dz − ∫ Ω η(z)|un|r dz < 1 n , ‖un‖ = 1. (4.8) We may assume that un w→ u in W 1,r 0 (Ω), un → u in Lr(Ω). (4.9) 16 Z. LIU, N. S. PAPAGEORGIOU EJDE-2023/30 Passing to the limit as n→∞ in (4.8) and using (4.9), we obtain∫ Ω a(z)|Du|r dz ≤ ∫ Ω η(z)|u|r dz ≤ λ̂a1(r)‖u‖rr, (4.10) ⇒ ∫ Ω a(z)|Du|r dz = λ̂a1(r)‖u‖rr (see (4.2)) (4.11) We claim u 6= 0. Otherwise we have ĉ‖Dun‖rr → 0 ⇒ un → 0 in W 1,r 0 (Ω), which contradicts that ‖un‖ = 1 for all n ∈ N. From (4.11) we see that we may assume that u ∈ intC+. Then from (4.10), we have∫ Ω a(z)|Du|r dz < λ̂a1(r)‖u‖rr, a contradiction, see (4.2). This completes the proof. � If r = 2 (the linear eigenvalue problem), then the spectral theorem for compact, self-adjoint operators, provides a complete description of the spectrum (−∆a, H1 0 (Ω)) which consists of a sequence {λ̂an(2)}n∈N ⊆ (0,∞) such that λ̂an(2)→∞. We denote by E(λ̂an(2)) the eigenspace corresponding to the eigenvalue λ̂an(2). We know that E(λ̂an(2)) is finite dimensional and E(λ̂an(2)) ⊆ C1 0 (Ω). (standard regularity theory). Moreover, the elements of E(λ̂an(2)) have the “Unique Continu- ation Property” (the UCP for short), that is, if u ∈ E(λ̂an(2)) and u(·) vanishes on a set of positive measure, then u ≡ 0 (see de Figueiredo-Gossez [3]). Let Hm = ⊕mi=1E(λ̂ai (2)), Ĥm+1 = H ⊥ m. Then we have H1 0 (Ω) = Ĥm ⊕ Ĥm+1. In this case we have variational characterizations for all the eigenvalues. So, we have λ̂a1(2) = inf [∫ Ω a(z)|Du|2 dz ‖u‖22 : u ∈ H1 0 (Ω), u 6= 0 ] (4.12) and for m ≥ 2 λ̂a1(2) = sup [∫ Ω a(z)|Du|2 dz ‖u‖22 : u ∈ Hm, u 6= 0 ] = inf [∫ Ω a(z)|Du|2 dz ‖u‖22 : u ∈ Ĥm, u 6= 0 ] . (4.13) In (4.12) and (4.13) the inf and sup are realized on the corresponding eigenspace E(λ̂am(2)). Proposition 4.2. If η ∈ L∞(Ω) and η(z) ≤ λ̂am(2) for a.a. z ∈ Ω, η 6≡ λ̂am(2), then there exists θ > 0 such that θ‖u‖r ≤ ∫ Ω a(z)|Du|2 dz − ∫ Ω η(z)|u|2 dz for all u ∈ Ĥm. Proof. If m = 1, then this follows from Proposition 4.1. So, assume m ≥ 2. Arguing by contradiction, suppose we can find {un}n∈N ⊆ Ĥm with ‖un‖ = 1 for all n ∈ N such that ∫ Ω a(z)|Dun|2 dz − ∫ Ω η(z)|un|2 dz < 1 n for all n ∈ N. (4.14) EJDE-2023/30 A WEIGHTED (p, 2)-EQUATION WITH DOUBLE RESONANCE 17 We can assume that un w→ u in H1 0 (Ω), un → u in L2(Ω). (4.15) If in (4.14) we pass to the limit as n→∞ and use (4.15), we obtain∫ Ω a(z)|Du|2 dz ≤ ∫ Ω η(z)|u|2 dz ≤ λ̂am(2)‖u‖22, (4.16) ⇒ ∫ Ω a(z)|Du|2 dz = λ̂am(2)‖u‖22 (since u ∈ Ĥm, see (4.13)) ⇒ u ∈ E(λ̂am(2)), u 6= 0. Then by the UCP we have u(z) 6= 0 for a.a. z ∈ Ω. Using this fact in (4.16), we obtain ∫ Ω a(z)|Du|2 dz < λ̂am(2)‖u‖22 which contradicts (4.13). This completes the proof. � Acknowledgments. The work was supported by NNSF of China Grant Nos. 12111530282, 12071413, and by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH. References [1] S. Aizicovici, N. S. Papageorgiou, V. 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Zhenhai Liu Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, China. Guangxi Colleges and Universities Key Laboratory of Optimization Control and Engi- neering Calculation, Guangxi Minzu University, Nanning, Guangxi, 530006, China Email address: zhhliu@hotmail.com Nikolaos S. Papageorgiou Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece Email address: npapg@math.ntua.gr 1. Introduction 2. Mathematical background and hypotheses 3. Three solution theorem 4. Appendix Acknowledgments References