Special Issue in honor of Alan C. Lazer Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 115–134. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu AN ELLIPTIC EQUATION INVOLVING THE SQUARE ROOT OF THE LAPLACIAN WITHOUT ASYMPTOTIC LIMITS YUTONG CHEN, JIABAO SU, MINGZHENG SUN, RUSHUN TIAN In memory of Professor Alan C. Lazer Abstract. In this article we show the existence of nontrivial solutions for nonlocal elliptic equations involving the square root of the Laplacian with the nonlinearity failing to have asymptotic limits at zero and at infinity. We use a combination of homotopy invariance of critical groups and the topological version of linking theorems. 1. Introduction This article concerns the nonlocal elliptic equation A1/2u = f(x, u), x ∈ Ω, u = 0, x ∈ ∂Ω, (1.1) where Ω is a smooth bounded domain of RN (N > 2) with Lipschitz boundary ∂Ω, and the nonlinearity f : Ω× R→ R is a Carathéodory function which satisfies the subcritical growth condition (A1) There exist C > 0 and 1 6 p < 2? := 2N N−1 such that |f(x, t)| 6 C(1 + |t|p−1) uniformly for a.e. x ∈ Ω and t ∈ R. (1.2) The nonlocal elliptic operator A1/2 in (1.1) is defined as the square root of the Laplacian −∆ in Ω with zero Dirichlet boundary data on ∂Ω. Let {λj , ϕj}∞j=1 satisfy −∆ϕj = λjϕj x ∈ Ω, ϕj = 0 x ∈ ∂Ω, (1.3) and ∫ Ω ϕjϕkdx = δj,k. For u ∈ H1 0 (Ω),write u(x) = ∑∞ j=1 αjϕj(x), x ∈ Ω, the nonlocal operator A1/2 appearing in (1.1) is defined (see [12]) as A1/2u :=∑∞ j=1 αj √ λjϕj . It has been proved in [12] that {µj := √ λj}∞j=1 are the eigen- values of A1/2 on Ω with the corresponding eigenfunctions {ϕj}∞j=1. The precise mathematical description and basic properties of the operator A1/2 will be stated in Section 2. 2010 Mathematics Subject Classification. 35A15, 35A16, 35R09, 35R11, 58E05. Key words and phrases. Square root of the Laplacian; homotopy invariance; critical groups; Morse theory; resonance; linking. c©2021 This work is licensed under a CC BY 4.0 license. Published October 6, 2021. 115 116 Y. CHEN, J. SU, M. SUN, R. TIAN EJDE/SI/01 In this article, we assume that f(x, 0) ≡ 0 so that (1.1) has a trivial solution u = 0. We will find via Morse theory nontrivial solutions to (1.1) in the situations the nonlinear term f has a linear growth and that there may not be the asymptotic limits of f(·, t)/t near both zero and infinity. We impose the following assumption on the nonlinearity f : (A1’) There exist p ∈ (2, 2?) and C > 0 such that for all s, t ∈ R, |f(x, s)− f(x, t)| 6 C(|s|p−2 + |t|p−2 + 1)|s− t|, uniformly for a.e. x ∈ Ω. (1.4) It is easy to see that (1.4) implies (1.2). Denote by 0 < µ1 < µ2 6 · · · 6 µk 6 · · · → ∞ the eigenvalues of A1/2. We impose on f the following conditions near zero and near infinity. (A2) There exist δ > 0 and k > 1 such that for two adjacent eigenvalues µk < µk+1 of A1/2, it holds that µkt 2 6 2F (x, t) 6 µk+1t 2, for |t| 6 δ, uniformly for a.e. x ∈ Ω. (1.5) (A3) There exist δ > 0 and k > 1 such that for two adjacent eigenvalues µk < µk+1 of A1/2, it holds that µk 6 f(x, t) t 6 µk+1, for 0 < |t| 6 δ, uniformly for a.e. x ∈ Ω. (1.6) (A4) There are ε > 0 and M > 0 such that for two adjacent eigenvalues µm < µm+1 of A1/2, it holds that µm + ε 6 f(x, t) t 6 µm+1 − ε, for |t| >M, uniformly for a.e. x ∈ Ω. (1.7) (A5) There are ε > 0 and M > 0 such that for two adjacent eigenvalues µm < µm+1 of A1/2, it holds that µm + ε 6 f(x, t) t 6 µm+1, 2F (x, t) 6 (µm+1 − ε)t2, (1.8) for |t| >M uniformly for a.e. x ∈ Ω. (A6) There are ε > 0 and M > 0 such that for two adjacent eigenvalues µm < µm+1 of A1/2, it holds that µm 6 f(x, t) t 6 µm+1 − ε, 2F (x, t) > (µm + ε)t2, (1.9) for |t| >M , uniformly for a.e. x ∈ Ω. Our main results are the following two theorems. Theorem 1.1. Assume (A1’). Then (1.1) admits at least one nontrivial weak solution in each of the following cases: (i) (A2), (A4) and µk 6= µm hold; (ii) (A2), (A5) and µk 6= µm hold. Theorem 1.2. Assume (A1’). Then (1.1) admits at least one nontrivial weak solution in each of the following cases: (i) (A3), (A4) and µk 6= µm hold; (ii) (A3), (A5) and µk 6= µm hold; (iii) (A3), (A6) and µk 6= µm hold. EJDE-2021/SI/01 AN ELLIPTIC EQUATION 117 Now we give some remarks on the conditions and conclusions. Conditions (A2) and (A3) were first introduced in [31], and (A4)–(A6) were first introduced in [29]. Conditions (A2) and (A3) mean that (1.1) may be resonant near zero between any two consecutive eigenvalues of A1/2, and there may not be any asymptotic limits of f(x, t)/t as t goes to zero. Obviously (A2) is weaker than (A3) and both of them are very general conditions when the trivial solution of (1.1) acts as a local saddle point. Condition (A4) means that (1.1) is non-resonant at infinity which contains lim|t|→∞ f(x, t)/t ∈ (µm, µm+1) (see [18, 48]) as a special case. Condition (A5) characterizes (1.1) as resonance near infinity at µm+1 from the left side, and (A6) characterizes (1.1) as resonance near infinity at µm from the right side. Semilinear variational problems with resonance have attracted much attention since the appearance on the great work [26] by Landesman and Lazer in 1970. In the setting of the semilinear elliptic equation −∆u = f(x, u) x ∈ Ω, u = 0 x ∈ ∂Ω, (1.10) one version of the Landesman-Lazer type resonance condition can be formulated as follows (see [1]), (LL) f(x, t)− λmt is bounded, and lim|t|→∞ ∫ t 0 (f(x, τ)− λmτ)dτ = ±∞. The crucial feature of the (LL) condition is the boundedness of f(x, t)−λmt which implies the asymptotic limit lim|t|→∞ f(x, t)/t = λm. In [34] the Saddle Point The- orem was applied to (1.10) with (LL) and infinite dimensional Morse theory was applied to (1.10) with (LL) (see [18, 19, 32, 48]). The study of the Landesman-Lazer type resonance problems motivated a large number of works involving resonance un- der different situations. Landesman, Robinson and Rumbos [27] considered (1.10) under a generalized Landesman-Lazer resonance condition, and Robinson [35] ex- tended the results in [27]. They treated the double resonance case in the sense that λm 6 lim inf |t|→∞ f(x, t) t 6 lim sup |t|→∞ f(x, t) t 6 λm+1 and multiple solutions were obtained via Leray-Schauder degree when the trivial solution was nondegenerate. Costa and Magalhães [24] treated (1.10) via the linking theorem under the so-called non-quadratic condition. Su and Tang [43] used via Morse theory and critical groups at infinity to study (1.10) with resonance and the nonlinearity g(x, t) := f(x, t)− λmt being unbounded and satisfying there exist c1 > 0, c2 > 0, θ ∈ (0, 1), and R > 0 such that g(x, t)t > 0 or g(x, t)t 6 0, c1|t|θ 6 |g(x, t)| 6 c2|t|θ, for all x ∈ Ω, |t| > R . (1.11) Su [44] studied (1.10) with the double resonance between two consecutive eigen- values λm and λm+1 and obtained multiple solutions via Morse theory and critical groups when 0 is a degenerate solution of (1.10). For other works involving (1.10) with various resonance we mention the works [1, 7, 8, 17, 27, 31, 35, 37, 38, 39, 40, 42] and their references. The existence of the asymptotic limits of f(x, t)/t near zero and near infinity had been required in most of the works mentioned above. Li, Per- era and Su[29] first treated the existence of nontrivial solutions of (1.10) without assuming asymptotic limits of f(x, t)/t near zero and infinity under the conditions in this article. However, the abstract homotopy theorem used in [29] should be modified. Therefore the analogue results for (1.10) are also new. 118 Y. CHEN, J. SU, M. SUN, R. TIAN EJDE/SI/01 The fractional powers of the Laplacian, such as the square root A1/2 of the Lapla- cian considered in this paper, appear in flame propagation and chemical reactions in liquids, population dynamics, geophysical fluid dynamics, anomalous diffusions in plasmas, and American options in finances (see [3, 25, 47]). In their well-known work [12], Cabré and Tan explored an essential characteristic of the nonlocal op- erator A1/2 in the sense that it could be realized in a local manner through the notion of harmonic extension and the Dirichlet-Neumann map due to Stein([41]) on Ω. More precisely, by introducing an harmonic extension problem in a cylinder C = Ω × (0,∞) in one more dimension, the nonlocal problem (1.1) is transformed to a local problem in the half cylinder C = Ω × (0,∞) with mixed boundary data which has a variational structure(see Section 2). Based on the variationl framework from [12], many efforts have been made in the applications of the variational and topological methods to (1.1) with various nonlinearities in getting the existence and multiplicity and many known results for (1.10) in literature have been extended to (1.1), see [2, 4, 6, 9, 10, 11, 12, 13, 14, 16, 20, 21, 36, 45, 46, 49] and some references therein. For examples, the existence of a positive solution of (1.1) for f(u) = |t|q−1t with 1 < q < N+1 N−1 was obtained in [12] by constrained minimization method, Tan studied in [45] the existence of a positive solution of (1.1) with critical nonlinear- ity case of f(t) = µt + |t| 2 N−1 t by the mountain pass theorem. In [23], nontrivial solutions and multiple solutions for (1.1) were obtained by comparing the critical groups at zero and infinity. To the authors’ knowledge, there are few works in literature for (1.1) with resonance near zero or near infinity at higher eigenvalues. Thus the results in this paper are quite new in the setting of the nonlocal problem considered here. This article is organized as follows. In Section 2 we present the functional frame- work related to (1.1) together with basic properties of the operator A1/2. Then we recall some preliminary results about Morse theory and critical groups. In Section 3 we compute the critical groups at zero, and in Section 4 we compute the critical groups at infinity. Finally in Section 5 we give the proofs of Theorems 1.1 and 1.2. 2. Preliminaries In this section we will give the preliminaries for the variational settings related to the nonlocal problem (1.1) and some abstract results in Morse theory. 2.1. Variational framework. We first recall briefly the functional framework of (1.1) given by Cabré and Tan [12]. Denote the half cylinder standing on Ω by C = {(x, y) : x ∈ Ω, y > 0} = Ω× (0,+∞) ⊂ RN+1 + and its lateral boundary by ∂LC = ∂Ω× (0,+∞). Consider the Sobolev space of functions with trace vanishing on ∂LC: H1 0,L(C) = { v ∈ L2(C) : v = 0 on ∂LC, ∫ C |∇v|2 dx dy <∞ } . Then H1 0,L(C) is a Hilbert space with the scalar product 〈v, w〉 = ∫ C ∇v∇w dxdy EJDE-2021/SI/01 AN ELLIPTIC EQUATION 119 and the norm ‖v‖ = (∫ C |∇v|2 dx dy )1/2 . From [12, Lemmas 2.4 and 2.5] we have the following embedding results. Proposition 2.1. The embedding from H1 0,L(C) into Lq(Ω) is continuous for all q ∈ [1, 2?] and is compact for all q ∈ [1, 2?). Moreover, there is cq > 0 such that(∫ Ω |v(x, 0)|qdx )1/q 6 cq (∫ C |∇v|2 dx dy )1/2 for all v ∈ H1 0,L(C). (2.1) We denote by trΩ the trace operator on Ω× {0} for functions in H1 0,L(C): trΩ v := v(·, 0), for v ∈ H1 0,L(C). Let V0(Ω) be the space of all traces on Ω× {0} of functions in H1 0,L(C), that is, V0(Ω) := { u = trΩ v : v ∈ H1 0,L(C) } . Then by [12, Lemma 2.10], V0(Ω) can be characterized as V0(Ω) = { u ∈ L2(Ω) : u = ∞∑ j=1 αjϕj satisfies ∞∑ j=1 α2 j √ λj < +∞ } (2.2) and the space H1 0,L(C) can be characterized as (see the proof of [12, Lemma 2.10]) H1 0,L(C) = { v ∈ L2(C) : v(x, y) = ∞∑ j=1 αjϕj exp(− √ λjy), ∞∑ j=1 α2 j √ λj < +∞ } . Where the pair {λj , ϕj}j∈N are the eigenvalue and the corresponding eigenfunction of −∆ on Ω with zero boundary value on ∂Ω, as stated in (1.3). For a given function u ∈ V0(Ω), its harmonic extension v to the cylinder C is the weak solution of the problem −∆v = 0 in C, v = 0 on ∂LC, v = u on Ω× {0}. (2.3) The idea of the harmonic extension was introduced in the pioneering work of Caf- farelli and Silvestre [15] where the fractional Laplacian in the whole space was considered. The definition and properties of the operator A1/2 are stated as follows. Proposition 2.2 ([12]). For u = ∑∞ j=1 αjϕj ∈ V0(Ω), there exists a unique har- monic extension v in C of u such that v ∈ H1 0,L(C), and it is given by the expansion v(x, y) = ∞∑ j=1 αjϕj(x) exp(− √ λjy), for all (x, y) ∈ C. (2.4) The operator A1/2 : V0(Ω)→ V∗0 (Ω) is given by the Dirichlet-Neumann map A1/2u := ∂v ∂ν ∣∣ Ω×{0}, (2.5) 120 Y. CHEN, J. SU, M. SUN, R. TIAN EJDE/SI/01 where V∗0 (Ω) is the dual space of V0(Ω) and where ν is the unit outer normal to C at Ω× {0}. We have A1/2u = ∞∑ j=1 αj √ λjϕj , (2.6) and that A1/2 ◦ A1/2 is equal to −∆ in Ω with zero Dirichlet boundary values on ∂Ω. The inverse A−1 1/2 is the unique positive square root of the inverse Laplacian (−∆)−1 in Ω with zero Dirichlet boundary values on ∂Ω. Now we consider the linear eigenvalue problem A1/2u = µu in Ω, u = 0 on ∂Ω. (2.7) By the definition of A1/2, we see that a nontrivial function u ∈ V0(Ω) is an eigen- function associated to the eigenvalue µ if and only if the harmonic extension v of u to the cylinder C satisfies −∆v = 0 in C, v = 0 on ∂LC, ∂v ∂ν = µu on Ω× {0}. (2.8) We have that { √ λj , ϕj}j∈N are the eigenvalues and the corresponding eigenfunc- tions of (2.7) (see [12, Lemma 2.13 ]). Setting µj = √ λj and ej(x, y) = ϕj(x) exp(−µjy) for all j ∈ N. (2.9) Then all the pairs {µj , ej}j∈N satisfy (2.8), the eigenfunction sequence {ej}j∈N forms an orthogonal basis of H1 0,L(C). The eigenvalue sequence {µj}j∈N has the following variational characterizations: µ1 = min v∈H1 0,L(C)\{0} ∫ C |∇v| 2 dx dy∫ Ω |v(x, 0)|2dx = ∫ C |∇e1|2 dx dy, (2.10) and µj = min v∈Pj\{0} ∫ C |∇v| 2 dx dy∫ Ω |v(x, 0)|2dx = ∫ C |∇ej |2 dx dy, (2.11) where Pj = {v ∈ H1 0,L(C) : 〈v, ei〉 = 0 for i = 1, 2, . . . , j − 1}. Moreover, µ1 is simple and 0 < µ1 < µ2 6 · · · 6 µj 6 · · · → ∞ as j →∞, and that each µj has finite multiplicity. For j ∈ N, let τj be the multiplicity of µj , that is µj−1 < µj = µj+1 = · · · = µj+τj−1 < µj+τj . Set H−(µj) = span{e1, . . . , ej−1}, H(µj) = span{ej , . . . , ej+τj−1}, H+(µj) = span{ej+τj , ej+τj+1, . . . , } = [ H−(µj)⊕H(µj) ]⊥ . Then H1 0,L(C) = H−(µj)⊕H(µj)⊕H+(µj). (2.12) EJDE-2021/SI/01 AN ELLIPTIC EQUATION 121 Proposition 2.3. The following variational inequalities hold:∫ C |∇v|2 dx dy 6 µj−1 ∫ Ω |v(x, 0)|2dx for all v ∈ H−(µj),∫ C |∇v|2 dx dy = µj ∫ Ω |v(x, 0)|2dx for all v ∈ H(µj),∫ C |∇v|2 dx dy > µj+`j ∫ Ω |v(x, 0)|2dx for all v ∈ H+(µj). We say that a function u ∈ V0(Ω) is a weak solution of (1.1) if the function v ∈ H1 0,L(C) with trΩ v = v(·, 0) = u weakly solves the extended problem −∆v = 0 in C, v = 0 on ∂LC, ∂v ∂ν = f(x, v(·, 0)) on Ω× {0}, (2.13) that is the function v satisfies the variational formula∫ C ∇v∇φdx dy = ∫ Ω f(x, v(x, 0))φ(x, 0)dx for all φ ∈ H1 0,L(C). (2.14) Since f satisfies (A1), it follows by Proposition 2.1 that the functional J (v) = 1 2 ∫ C |∇v|2 dx dy − ∫ Ω F (x, v(x, 0))dx, v ∈ H1 0,L(C) (2.15) is well-defined on H1 0,L(C) and is of class C1 with derivative given by 〈J ′(v), φ〉 = ∫ C ∇v∇φdx dy − ∫ Ω f(x, v(x, 0))φ(x, 0)dx. (2.16) Therefore critical points of J are exactly weak solutions of (2.13) and then the traces of critical points of J are exactly weak solutions of (1.1). We will apply Morse theory and critical groups to find critical points of J and the following results will be necessary. Proposition 2.4. Assume that (A1’) holds. Then J ∈ C2−0(H1 0,L(C),R). Proof. The arguments are similar to that in [5] and we sketch out them for the readers’ convenience. We only need to prove I(v) = ∫ Ω F (x, v(x, 0))dx is C2−0 on H1 0,L(C). For any v, w, φ ∈ H1 0,L(C) with ‖φ‖ = 1, we have by (A1’), Proposition 2.1 and Hölder inequality that |〈I ′(v)− I ′(w), φ〉| 6 (∫ Ω |f(x, v(x, 0))− f(x,w(x, 0))| p p−1 dx ) p−1 p (∫ Ω |φ(x, 0)|pdx )1/p 6 C (∫ Ω |v(x, 0)− w(x, 0)| p p−1 ( 1 + |v(x, 0)|p−2 + |w(x, 0)|p−2 ) p p−1 dx ) p−1 p 6 C (∫ Ω |v(x, 0)− w(x, 0)|pdx )1/p × (∫ Ω ( 1 + |v(x, 0)|p−2 + |w(x, 0)|p−2 ) p p−2 dx ) p−2 p 6 C (1 + ‖v‖+ ‖w‖)p−2 ‖v − w‖. (2.17) 122 Y. CHEN, J. SU, M. SUN, R. TIAN EJDE/SI/01 For ς > 0 and ‖v‖ 6 ς, ‖w‖ 6 ς, it follows from (2.17) that ‖I ′(v)− I ′(w)‖ = sup φ∈H1 0,L(C),‖φ‖=1 |〈I ′(v)− I ′(w), φ〉| 6 C(ς)‖v − w‖, where C(ς) is a constant depending on ς. Therefore I ′ is locally Lipschitz continu- ous. � Proposition 2.5 ([23, Lemma 3.1]). Assume that (A1) holds. Then any a bounded sequence {vn} ⊂ H1 0,L(C) satisfying J ′(vn)→ 0 as n→∞ has a convergent subse- quence. 2.2. Preliminaries about Morse theory. In this subsection we collect some abstract results on Morse theory [18, 32] for a C1 functional defined on a Banach space X. Let J ∈ C1(X,R) and K = {v ∈ X : J ′(v) = 0}. For c ∈ R we denote J c = {v ∈ X : J (v) 6 c} and Kc = K ∩ {v ∈ X : J (v) = c}. We say that J satisfies the Palais-Smale condition at the level c ∈ R if any sequence {vn} ⊂ X satisfying J (vn) → c and J ′(vn) → 0 as n → ∞ has a convergent subsequence. We say that J satisfies the Palais-Smale condition if J satisfies the Palais-Smale condition at each c ∈ R. Let v0 be an isolated critical point of J with J (v0) = c ∈ R, and U be a neigh- borhood of v0 such that U ∩ Kc = {v0}. The group Cq(J , v0) := Hq(J c ∩ U,J c ∩ U \{v0}), q ∈ Z is called the q-th critical group of J at v0, where H∗(A,B) denotes a singular relative homology group of the pair (A,B) with integer coefficients (see [18, 32]). Assume that J (K) is bounded from below by a ∈ R and J satisfies the Palais- Smale condition at all c 6 a. The group Cq(J ,∞) := Hq(X,J a), q ∈ Z, is called the q-th critical group of J at infinity([8]). Assume that J satisfies the Palais-Smale condition and K is a finite set con- taining 0. Then the critical groups of J at infinity and at 0 are well-defined. The basic idea of Morse theory tells us that if K = {0} then Cq(J ,∞) ∼= Cq(J , 0) for all q ∈ Z. It follows that if Cq(J ,∞) 6∼= Cq(J , 0) for some q ∈ Z then J must have a nontrivial critical point. Therefore the basic method in applying Morse theory to find nontrivial critical points of J is to compute critical groups Cq(J , 0) and Cq(J ,∞). The groups Cq(J , 0) can be computed partially when J has a local linking structure at zero. Proposition 2.6 ([30]). Let J ∈ C1(X,R) satisfy the Palais-Smale condition and 0 ∈ K. Assume that J has a local linking structure at 0 with respect to X = X−0 ⊕X + 0 , i.e. there exists ρ > 0 such that J (v) > 0 for v ∈ X+ 0 , 0 < ‖v‖ 6 ρ, J (v) 6 0 for v ∈ X−0 , ‖v‖ 6 ρ. (2.18) Then C`0(J , 0) 6∼= 0 if `0 = dimX−0 <∞. The concept of local linking in Proposition 2.6 was introduced by Li and Liu [28] for the existence of nontrivial critical points. If X is a Hilbert space and J is of C2 then Cq(J , 0) can be computed clearly provided `0 is the Morse index or augmented Morse index of J at 0. See [44, Proposition 2.3]. Proposition 2.7 ([8]). Assume X = X1 ⊕ X2 and J ∈ C1(X,R) satisfies the Palais-Smale condition. If J is bounded from below on X2 and is anti-coercive EJDE-2021/SI/01 AN ELLIPTIC EQUATION 123 on X1, i.e. J (v) → −∞ as v ∈ X1 with ‖v‖ → ∞, then C`(J ,∞) 6∼= 0 if ` = dimX1 <∞. The above proposition is a version of the famous Rabinowitz’s Saddle Point Theorem [34, Theorem 4.6]. We regard Propositions 2.6 and 2.7 as the topologi- cal versions of corresponding linking theorems since there are no minimax values involved. Next we give two theorems about the homotopy invariance of critical groups that can be used to compute directly the critical groups at isolated critical point and infinity respectively. Theorem 2.8 ([18, 32]). Let X be a Hilbert space and {Jt ∈ C2−0(X,R) : t ∈ [0, 1]} be a family of functional satisfying the Palais-Smale condition. Assume that there exists an open set U such that Jt has a unique critical point vt ∈ U for each t ∈ [0, 1] and t 7→ Jt is continuous in C1(Ū) topology. Then Cq(Jt, vt) is independent of t ∈ [0, 1]. Theorem 2.9 ([22]). Let X be a Hilbert space and let Jt ∈ C1(X,R) be a family of functionals, t ∈ [0, 1]. Assume that each Jt satisfies the Palais-Smale condition, J ′t and ∂tJt are locally Lipschitz continuous in u. If there exists a ∈ R and δ > 0 such that for some C > 0 Jt(u) 6 a ⇒ ‖∂tJt(u)‖ 6 C‖u‖2, for all t ∈ [0, 1], (2.19) Jt(u) 6 a ⇒ ‖J ′t (u)‖ > δ‖u‖, for all t ∈ [0, 1], (2.20) then Cq(J0,∞) ∼= Cq(J1,∞). (2.21) We point out that Theorem 2.9 is a new modification of [29, Theorem 3.1] (see [33]), where the given conditions were not sufficient to ensure the existence of flow. This new version of homotopy theorem has its own meanings and can be applied to many variational problems. The proof of Theorem 2.9 has been given in [22] where another type of nonlocal variational problem was studied. 3. Critical groups at zero In this section we compute C∗(J , 0) under the assumptions (A2) and (A3). We make a conventional assumption that the trivial solution 0 of (1.1) is isolated. By Proposition 2.5 we see that J satisfies the Palais-Smale condition over any a closed ball centered at 0. We use the following orthogonal decomposition: H1 0,L(C) = H−(µk)⊕H(µk)⊕H+(µk), Hk = ⊕µj6µk H(µj), `k = dimHk. (3.1) Proposition 3.1. Assume that (A1) and (A2) hold. Then C`k(J , 0) 6= 0. Proof. We will show that J has a local linking structure at 0 with respect to H1 0,L(C) = Hk ⊕H⊥k . (i) Since Hk is finite dimensional, by (A2), Propositions 2.1 and 2.3 we can find ρ > 0 small such that for all v ∈ Hk with ‖v‖ 6 ρ, J (v) 6 − ∫ Ω ( F (x, v(x, 0))− 1 2 µk|v(x, 0)|2 ) dx 6 0. (3.2) 124 Y. CHEN, J. SU, M. SUN, R. TIAN EJDE/SI/01 (ii) For v ∈ H⊥k , we write v = z + w, where z ∈ H(µk+1), w ∈ H⊥k+1. By Proposition 2.3 we have J (v) > 1 2 ( 1− µk+1 µk+2 ) ‖w‖2 − ∫ Ω ( F (x, v(x, 0))− 1 2 µk+1|v(x, 0)|2 ) dx. (3.3) For |v(x, 0)| 6 δ, by (A2) we have∫ {|v(x,0)|6δ} ( F (x, v(x, 0))− 1 2 µk+1|v(x, 0)|2 ) dx 6 0. (3.4) It follows from (A1) and Proposition 2.1 that for some q ∈ (2, 2?],∫ {|v(x,0)|>δ} ( F (x, v(x, 0))− 1 2 µk+1|v(x, 0)|2 ) dx 6 C‖w‖q. (3.5) Hence by (3.3)–(3.5) we have J (v) > 1 2 ( 1− µk+1 µk+2 ) ‖w‖2 − C‖w‖q. (3.6) Since q > 2, it follows from (3.6) that there is ρ > 0 small such that J (v) > 0, ∀ 0 < ‖v‖ 6 ρ with w 6= 0. (3.7) We can choose ρ > 0 so small that ‖v‖ 6 ρ⇒ ‖z‖ 6 ρ⇒ |z(x, 0)| 6 δ for all x ∈ Ω. Then by (A2) we have F (x, z(x, 0))− 1 2 µk+1z 2(x, 0) 6 0 uniformly in x ∈ Ω. Thus J (z) = − ∫ Ω ( F (x, z(x, 0))− 1 2 µk+1z 2(x, 0) ) dx > 0. Let z∗ ∈ H(µk+1) be such that 0 < ‖z∗‖ 6 ρ and J (z∗) = 0. Then F (x, z∗(x, 0))− 1 2 µk+1z 2 ∗(x, 0) = 0 uniformly in x ∈ Ω, and so f(x, z∗(x, 0)) = µk+1z∗(x, 0) uniformly in x ∈ Ω. As z∗ ∈ H(µk+1), going back to (2.13), we see that z∗ is a nontrivial solution for (2.13) and z∗(·, 0) is a solution of (1.1). We conclude that there is ρ > 0 small such that for all ‖v‖ 6 ρ with w = 0 and z 6= 0, J (v) = J (z) = − ∫ Ω ( F (x, z(x, 0))− 1 2 µk+1z 2(x, 0) ) dx > 0. (3.8) Otherwise, for any ε > 0 there exists 0 6= zε ∈ H(µk+1) such that ‖zε‖ < ε and J (zε) = 0. Then zε(·, 0) is a nontrivial solution of (1.1). It contradicts the isolation of the trivial solution. It follows from (3.7) and (3.8) that J (v) > 0, for all v ∈ H⊥k with 0 < ‖v‖ 6 ρ. Therefore J has a local linking structure at 0 with respect to H1 0,L(C) = Hk⊕H⊥k . Since `k = dimHk <∞, it follows from Proposition 2.6 that C`k(J , 0) 6= 0. � Proposition 3.2. Assume that (A1’) and (A3) hold. Then Cq(J , 0) = δq,`kF. EJDE-2021/SI/01 AN ELLIPTIC EQUATION 125 Proof. For v ∈ H1 0,L(C), we write v = z + φ + w and set v̂ = −z + φ + w, where z ∈ Hk, φ ∈ H(µk+1) and w ∈ H⊥k+1. We define a family of functionals Jt(v) = (1− t)J (v) + t 2 ( −‖z‖2 + ‖φ‖2 + ‖w‖2 ) , t ∈ [0, 1]. (3.9) By (A1’) and Proposition 2.4 we have that Jt ∈ C2−0(H1 0,L(C),R) and 〈J ′t (v), ϕ〉 = (1− t)〈J ′(v), ϕ〉+ t〈v̂, ϕ〉. (3.10) Now we show that there is ρ > 0 such that v = 0 is a unique critical point of Jt in the ball Bρ(0) for all t ∈ [0, 1]. Denote g(x, t) = f(x, t)− µk+1t. Then by (A3) we have that 0 < −g(x, t) t 6 µk+1 − µk, 0 < |t| 6 δ. Then for |v(x, 0)| 6 δ, g(x, v(x, 0))v̂(x, 0) 6 { 0, if v(x, 0)ṽ(x, 0) > 0, (µk+1 − µk)z2(x, 0), if v(x, 0)ṽ(x, 0) < 0. (3.11) Hence ∫ {|v(x,0)|6δ} g(x, v(x, 0))v̂(x, 0)dx 6 (µk+1 − µk) ∫ Ω z2(x, 0)dx. (3.12) Since trΩHk and trΩH(µk+1) are finite dimensional, there is a ρ > 0 such that ‖z‖ 6 ρ⇒ |z(x, 0)| 6 δ 3 , ‖φ‖ 6 ρ⇒ |φ(x, 0)| 6 δ 3 . For ‖v‖ 6 ρ and |v(x, 0)| > δ, |v(x, 0)| 6 |w(x, 0)|+ |φ(x, 0)|+ |z(x, 0)| 6 |w(x, 0)|+ 2 3 δ, and so |v(x, 0)| < 3|w(x, 0)|, |v̂(x, 0)| < 3|w(x, 0)|. Thus by (A1’) we have∫ {|v(x,0)|>δ} |g(x, v(x, 0))v̂(x, 0)|dx 6 C ∫ {|v(x,0)|>δ} |v(x, 0)|p−1|v̂(x, 0)|dx 6 C ∫ {|v(x,0)|>δ} |w(x, 0)|pdx 6 C‖w‖p. (3.13) Now for ‖v‖ 6 ρ, it follows from (3.12) and (3.13) that 〈J ′(v), v̂〉 = 〈v, v̂〉 − µk+1 ∫ Ω vv̂dx− ∫ Ω g(x, v(x, 0))v̂(x, 0)dx > ( 1− µk+1 µk+2 ) ‖w‖2 − ( ‖z‖2 − µk+1 ∫ Ω |z(x, 0)|2dx ) − ∫ {|v(x,0)|6δ} g(x, v(x, 0))v̂(x, 0)dx− ∫ {|v(x,0)|>δ} g(x, v(x, 0))v̂(x, 0)dx > ( 1− µk+1 µk+2 ) ‖w‖2 − ( ‖z‖2 − µk ∫ Ω |z(x, 0)|2dx ) − C‖w‖p. (3.14) 126 Y. CHEN, J. SU, M. SUN, R. TIAN EJDE/SI/01 Therefore for ‖v‖ 6 ρ, take ϕ = v̂ in (3.10), we obtain from (3.14) that 〈J ′t (v), v̂〉 > (1− t) [( 1− µk+1 µk+2 ) ‖w‖2 − ( ‖z‖2 − µk ∫ Ω |z(x, 0)|2dx )] − (1− t)C‖w‖p + t‖v‖2. (3.15) Since p > 2, it follows that 0 is the only critical point of Jt in Bρ(0) for all t ∈ [0, 1] if ρ > 0 is sufficiently small. Since J1(v) = 1 2 ( −‖z‖2 + ‖φ‖2 + ‖w‖2 ) (3.16) is a C2 functional and has 0 as a non-degenerate critical point with Morse index `k = dimHk, it follows that Cq(J1, 0) = δq,`kF, ∀q ∈ Z. (3.17) By Theorem 2.8 and (3.17) we have Cq(J , 0) = Cq(J0, 0) ∼= Cq(J1, 0) = δq,`kF. (3.18) The proof is complete. � 4. Critical groups at infinity In this section we compute C∗(J ,∞) under the corresponding assumptions (A4)– (A6). We use the following orthogonal decomposition: H1 0,L(C) = H−(µm)⊕H(µm)⊕H+(µm), Hm = ⊕µj6µmH(µj), `m = dimHm. (4.1) We will use C to denote various positive constants in the sequel. Proposition 4.1. Assume that (A1’) and (A4) hold. Then J satisfies the Palais- Smale condition and Cq(J ,∞) ∼= δq,`mF. Proof. We will apply Theorem 2.9 to prove this proposition. Set f̃(x, t) = f(x, t)− (µm+1 − ε)t, F̃ (x, t) = ∫ t 0 f̃(x, ζ)dζ. Then J can be rewritten as J (v) = 1 2 ∫ C |∇v|2 dx dy − 1 2 (µm+1 − ε) ∫ Ω |v(x, 0)|2dx− ∫ Ω F̃ (x, v(x, 0))dx. For v ∈ H1 0,L(C), we write v = z + w and set ṽ = −z + w where z ∈ Hm and w ∈ H⊥m. We define a family of functionals Jt(v) = (1− t)J (v) + t 2 ( −‖z‖2 + ‖w‖2 ) , t ∈ [0, 1]. (4.2) By (A1’) we have that Jt ∈ C2−0(H1 0,L(C),R) and 〈J ′t (v), ϕ〉 = (1− t)〈J ′(v), ϕ〉+ t〈ṽ, ϕ〉, ∀v, ϕ ∈ H1 0,L(C). (4.3) By (A4) we have 0 6 − f̃(x, t) t 6 µm+1 − µm − 2ε, ∀|t| >M, x ∈ Ω. EJDE-2021/SI/01 AN ELLIPTIC EQUATION 127 For |v(x, 0)| >M we have that f̃(x, v(x, 0))ṽ(x, 0) 6 { 0, v(x, 0)ṽ(x, 0) > 0, (µm+1 − µm − 2ε)z2(x, 0), v(x, 0)ṽ(x, 0) < 0. Hence∫ {|v(x,0)|>M} f̃(x, v(x, 0))ṽ(x, 0)dx 6 (µm+1 − µm − 2ε) ∫ Ω |z(x, 0)|2dx. (4.4) By (A1) there is C > 0 such that∫ {|v(x,0)| ε µm+1 ‖w‖2 − [ ‖z‖2 − (µm+1 − ε) ∫ Ω |z(x, 0)|2dx ] − (∫ {|v(x,0)|M} ) f̃(x, v(x, 0))ṽ(x, 0)dx > ε µm+1 ‖w‖2 − [ ‖z‖2 − (µm + ε) ∫ Ω |z(x, 0)|2dx ] − C‖ṽ‖ > ε µm+1 ‖w‖2 + ε µm ‖z‖2 − C‖ṽ‖ > ε µm+1 ‖v‖2 − C‖v‖. (4.6) Taking ϕ = ṽ in (4.3), we obtain from (4.6) that 〈J ′t (v), ṽ〉 > (1− t) [ ε µm+1 ‖v‖2 − C‖v‖ ] + t‖ṽ‖2 > Cε‖v‖2 − C‖v‖. (4.7) where Cε = min{1, ε/µm+1}. By (4.7) we see that any a Palais-Smale sequence of Jt must be bounded. By Proposition 2.5([23, Lemma 3.1]) we have that Jt satisfies the Palais-Smale condition for all t ∈ [0, 1]. Moreover, it follows from (4.7) that there are a� −1 and δ > 0 such that Jt(v) 6 a⇒ ‖J ′t (v)‖ > δ‖v‖. (4.8) For any a < −1 being fixed, it always holds that Jt(v) 6 a⇒ |∂tJt(v)| 6 C‖v‖2. (4.9) From the definition we see that J0(v) = J (v) and J1(v) = 1 2 (−‖z‖2 + ‖w‖2). (4.10) Then J1 is a C2 functional and has 0 as a unique non-degenerate critical point with Morse index `m = dimHm. It follows that Cq(J1,∞) = Cq(J1, 0) ∼= δq,`mF, ∀q ∈ Z. (4.11) 128 Y. CHEN, J. SU, M. SUN, R. TIAN EJDE/SI/01 By Theorem 2.9 and (4.11) we have Cq(J ,∞) = Cq(J0,∞) ∼= Cq(J1,∞) = δq,`mF. (4.12) The proof is complete. � We note here that the conclusion of Proposition 4.1 is valid in the case that (A1) holds and lim|t|→∞ f(x,t) t = ξ ∈ (µm, µm+1). This is the completely non-resonant case at infinity. See [18] for a proof in abstract version. However, the arguments in [18] would not be applied to the case of Proposition 4.1. The next two results involve with (1.1) being slight resonant near infinity from one side of an eigenvalue of A1/2. Proposition 4.2. Assume that (A1’) and (A5) hold. Then J satisfies the Palais- Smale condition and Cq(J ,∞) = δq,`mF. Proof. We set f̃(x, t) = f(x, t)− µm+1t, F̃ (x, t) = ∫ t 0 f̃(x, ζ)dζ and rewrite J as J (v) = 1 2 ‖v‖2 − 1 2 µm+1 ∫ Ω |v(x, 0)|2dx− ∫ Ω F̃ (x, v(x, 0))dx. For v ∈ H1 0,L(C), we write v = z + φ+w, where z ∈ Hm, φ ∈ H(µm+1), w ∈ H⊥m+1 and set ṽ = −z + φ+ w. Define a family of functionals Jt(v) = (1− t)J (v) + t 2 (−‖z‖2 + ‖φ‖2 + ‖w‖2), t ∈ [0, 1]. (4.13) By (A1’) we have that Jt ∈ C2−0(H1 0,L(C),R) and 〈J ′t (v), ϕ〉 = (1− t)〈J ′(v), ϕ〉+ t 〈ṽ, ϕ〉 . (4.14) By (A5) we have that 0 6 − f̃(x, t) t 6 µm+1 − µm − ε, ∀|t| >M, x ∈ Ω. Thus for |v(x, 0)| >M we have f̃(x, v(x, 0))ṽ(x, 0) 6 { 0, v(x, 0)ṽ(x, 0) > 0, (µm+1 − µm − ε)z2(x, 0), v(x, 0)ṽ(x, 0) < 0. (4.15) Hence∫ {|v(x,0)|>M} f̃(x, v(x, 0))ṽ(x, 0)dx 6 (µm+1 − µm − ε) ∫ Ω |z(x, 0)|2dx, (4.16) and there is C > 0 such that∫ {|v(x,0)| ( 1− µm+1 µm+2 ) ‖w‖2 − [ ‖z‖2 − µm+1 ∫ Ω |z(x, 0)|2dx ] − (∫ {|v(x,0)|M} ) f̃(x, v(x, 0))ṽ(x, 0)dx > ( 1− µm+1 µm+2 ) ‖w‖2 − [ ‖z‖2 − µm+1 ∫ Ω |z(x, 0)|2dx ] − ∫ Ω (µm+1 − µm − ε)|z(x, 0)|2dx− C‖ṽ‖ > ( 1− µm+1 µm+2 ) ‖w‖2 + ε µm ‖z‖2 − C‖ṽ‖. (4.18) Taking ϕ = ṽ in (4.14), then we obtain from (4.18) that 〈J ′t (v), ṽ〉 > (1− t) [( 1− µm+1 µm+2 ) ‖w‖2 + ε µm ‖z‖2 − C‖ṽ‖ ] + t‖ṽ‖2. (4.19) We prove that there exists δ > 0 such that for any a ∈ R fixed Jt(v) 6 a⇒ ‖J ′t (v)‖ > δ‖v‖. (4.20) Arguing by contradiction, we assume that there exists tn ∈ [0, 1], vn ∈ H1 0,L(C) such that Jtn(vn)→ −∞ and ‖J ′tn(vn)‖ < 1 n ‖vn‖, (4.21) this means that ‖vn‖ → ∞, as n→∞. (4.22) We denote v̂n = vn ‖vn‖ = ẑn + φ̂n + ŵn. Then ‖v̂n‖ ≡ 1. It follows from (4.21) that 〈J ′tn(vn), ṽn〉 ‖vn‖2 → 0, as n→∞. (4.23) We have by (4.19) that 〈J ′tn(vn), ṽn〉 ‖vn‖2 > (1− tn) [( 1− µm+1 µm+2 ) ‖ŵn‖2 + ε µm ‖ẑn‖2 − C ‖ṽn‖ ] + tn. (4.24) Since tn ∈ [0, 1], ‖ŵn‖2 6 1 and ‖ẑn‖2 6 1, we may assume, up to a subsequence, that tn → t∗ ∈ [0, 1], ‖ẑn‖2 → α ∈ [0, 1], ‖ŵn‖2 → β ∈ [0, 1], n→∞. (4.25) It follows from (4.23)–(4.25) that (1− t∗) [( 1− µm+1 µm+2 ) β + ε µm α ] + t∗ 6 0. (4.26) It must be that t∗ = 0 and α = β = 0. This means that ẑn → 0, ŵn → 0, as n→∞. 130 Y. CHEN, J. SU, M. SUN, R. TIAN EJDE/SI/01 Since ‖v̂n‖ ≡ 1, it holds that φ̂n → φ̂ 6= 0, ‖φ̂‖ = 1. It follows that Jtn(vn) = (1− tn) (1 2 ‖vn‖2 − ∫ Ω F (x, vn(x, 0))dx ) + tn 2 (−‖zn‖2 + ‖φn‖2 + ‖wn‖2) > (1− tn)‖vn‖2 (1 2 ε µm+1 ‖φ̂n‖2 − C(‖ẑn‖2 + ‖ŵn‖2) ) − C + 1 2 tn‖vn‖2(−‖ẑn‖2 + ‖φ̂n‖2 + ‖ŵn‖2)→∞, n→∞. (4.27) This proves (4.20). Using the same arguments above we can show that for each t ∈ [0, 1], Jt satisfies the Palais-Smale condition. Moreover, it is easy to see that for any a < −1 being fixed, it always holds that Jt(v) 6 a⇒ |∂tJt(v)| 6 C‖v‖2. (4.28) Since J1(v) = 1 2 (−‖z‖2 + ‖φ‖2 + ‖w‖2) (4.29) is a C2 functional and has 0 as a unique non-degenerate critical point with Morse index `m = dimHm, it follows that Cq(J1,∞) = Cq(J1, 0) ∼= δq,`mF, ∀q ∈ Z. (4.30) By (4.20), (4.28), Theorem 2.9 and (4.30) we have Cq(J ,∞) = Cq(J0,∞) ∼= Cq(J1,∞) = δq,`mF. (4.31) The proof is complete. � Proposition 4.3. Assume that (A1) and (A6) hold. Then J satisfies the Palais- Smale condition and C`m(J ,∞) � 0. Proof. We will apply Proposition 2.7. We first prove that J satisfies the Palais- Smale condition. Although the argument is somewhat similar to that of the previous proposition, we prefer to give the details. Assume that {vn} ⊂ H1 0,L(C) satisfies |J (vn)| 6 C, J ′(vn)→ 0, as n→∞. (4.32) By Proposition 2.5, we only need to prove that {vn} is bounded. Assume that ‖vn‖ → ∞ as n → ∞. Set v̂n = vn ‖vn‖ = ẑn + φ̂n + ŵn where ẑn ∈ Hm−1, φ̂n ∈ H(µm) and ŵn ∈ H⊥m. Then ‖v̂n‖ ≡ 1. Set f̃(x, t) = f(x, t)− µmt, F̃ (x, t) = ∫ t 0 f̃(x, ζ)dζ. By (A6) we have 0 6 f̃(x, t) t 6 µm+1 − µm − ε, ∀|t| >M, x ∈ Ω. For v ∈ H1 0,L(C), set ṽ = −(z + φ) + w. Then for |v(x, 0)| >M , we have f̃(x, v(x, 0))ṽ(x, 0) 6 { 0, v(x, 0)ṽ(x, 0) < 0, (µm+1 − µm − ε)w2(x, 0), v(x, 0)ṽ(x, 0) > 0. (4.33) EJDE-2021/SI/01 AN ELLIPTIC EQUATION 131 Hence∫ {|v(x,0)|>M} f̃(x, v(x, 0))ṽ(x, 0)dx 6 (µm+1 − µm − ε) ∫ Ω |w(x, 0)|2dx. (4.34) There is C > 0 such that∫ {|v(x,0)|M} ) f̃(x, v(x, 0))ṽ(x, 0)dx > ( ‖w‖2 − µm ∫ Ω |w(x, 0)|2dx ) − ( ‖z‖2 − µm ∫ Ω |z(x, 0)|2dx ) − ∫ Ω (µm+1 − µm − ε)|w(x, 0)|2dx− C‖ṽ‖ > ( µm µm−1 − 1 ) ‖z‖2 + ε µm+1 ‖w‖2 − C‖ṽ‖. (4.36) By (4.32) and (4.36), we obtain o(‖vn‖) = 〈J ′(vn), ṽn〉 > ( µm µm−1 − 1 ) ‖zn‖2 + ε µm+1 ‖wn‖2 − C‖ṽn‖. (4.37) Therefore, o(1) ‖vn‖ > ( µm µm−1 − 1 ) ‖ẑn‖2 + ε µm+1 ‖ŵn‖2 − C ‖vn‖ . (4.38) Since ‖ŵn‖2 6 1 and ‖ẑn‖2 6 1, we assume, up to a subsequence, that ‖ẑn‖2 → α ∈ [0, 1], ‖ŵn‖2 → β ∈ [0, 1], n→∞. (4.39) It follows from (4.38) that( µm µm−1 − 1 ) α+ ε µm+1 β 6 0. (4.40) It must be that α = β = 0 and thus ẑn → 0 and ŵn → 0 as n→∞. Since ‖v̂n‖ ≡ 1, it follows that φ̂n → φ̂ 6= 0, ‖φ̂‖ = 1. Now we have J (vn) = 1 2 ‖vn‖2 − ∫ Ω F (x, vn(x, 0))dx 6 ‖vn‖2 ( − 1 2 ε µm ‖φ̂n‖2 + C(‖ẑn‖2 + ‖ŵn‖2) ) + C → −∞ (4.41) as n→∞. This contradicts (4.32). 132 Y. CHEN, J. SU, M. SUN, R. TIAN EJDE/SI/01 Next we prove that J satisfies the geometrical assumptions of Proposition 2.7 with respect to H1 0,L(C) = Hm ⊕H⊥m. From (A6) it follows that (µm + ε)t2 − C 6 2F (x, t) 6 (µm+1 − ε)t2 + C (4.42) for some C > 0. Then for w ∈ H⊥m, J (w) > 1 2 ‖w‖2 − 1 2 ∫ Ω (µm+1 − ε)|w(x, 0)|2dx− C > ε 2 ∫ Ω |w(x, 0)|2dx− C > −C. (4.43) For z ∈ Hm, J (z) 6 1 2 ‖z‖2 − 1 2 ∫ Ω (µm + ε)|z(x, 0)|2dx+ C 6 − ε 2µm ‖z‖2 + C → −∞, ‖z‖ → ∞. (4.44) As dimHm = `m <∞, we have by Proposition 2.7 that C`m(J ,∞) � 0. The proof is complete. � 5. Proofs of main theorems Proof of Theorem 1.1. (i) By Proposition 3.1, we have that C`k(J , 0) 6= 0. By Proposition 4.1, J satisfies the Palais-Smale condition and Cq(J ,∞) = δq,`mF. Since µk 6= µm implies `k 6= `m, it follows that C`k(J ,∞) � C`k(J , 0). Therefore J has at least one nontrivial critical point. The case (ii) is proved in a similar way. � Proof of Theorem 1.2. (iii) By Proposition 3.2, we have that Cq(J , 0) = δq,`kF. 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Sinica, N.S., 5 (1989), 101–113. [49] X. Yu; The Nehari manifold for elliptic equation involving the square root of the Laplacian. J. Differential Equations, 252 (2012), 1283–1308. Yutong Chen School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Email address: chenyutong@cnu.edu.cn Jiabao Su School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Email address: sujb@cnu.edu.cn Mingzheng Sun College of Sciences, North China University of Technology, Beijing 100144, China Email address: suncut@163.com Rushun Tian School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Email address: rushun.tian@cnu.edu.cn 1. Introduction 2. Preliminaries 2.1. Variational framework 2.2. Preliminaries about Morse theory. 3. Critical groups at zero 4. Critical groups at infinity 5. Proofs of main theorems Acknowledgments References