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BY 4.0 license Open Access Published by De Gruyter November 4, 2020

Existence and Concentration of Solutions for Choquard Equations with Steep Potential Well and Doubly Critical Exponents

  • Yong-Yong Li , Gui-Dong Li and Chun-Lei Tang EMAIL logo

Abstract

In this paper, we investigate the non-autonomous Choquard equation

- Δ u + λ V ( x ) u = ( I α F ( u ) ) F ( u ) in R N ,

where N4, λ>0, VC(RN,R) is bounded from below and has a potential well, Iα is the Riesz potential of order α(0,N) and F(u)=12α*|u|2α*+12*α|u|2*α, in which 2α*=N+αN-2 and 2*α=N+αN are upper and lower critical exponents due to the Hardy–Littlewood–Sobolev inequality, respectively. Based on the variational methods, by combining the mountain pass theorem and Nehari manifold, we obtain the existence and concentration of positive ground state solutions for 𝜆 large enough if 𝑉 is nonnegative in RN; further, by the linking theorem, we prove the existence of nontrivial solutions for 𝜆 large enough if 𝑉 changes sign in RN.

MSC 2010: 35J15; 35B33; 35J20; 35D30

1 Introduction and Main Results

In this paper, we devote ourselves to studying the nonlinear Choquard type equation

(1.1)-Δu+V(x)u=μ(IαG(x,u))g(x,u)inRN,

where N3, μ(0,+), Iα is the Riesz potential of order α(0,N) defined for all xRN\{0} by

Iα(x)=Aα|x|N-α=Γ(N-α2)Γ(α2)2απN2|x|N-α

with Γ denoting the Gamma function, ∗ is the convolution, gC(RN×R,R) and G(x,t)=0tg(x,s)ds.

The Choquard equation (1.1) appears as classical models in several physical contexts. Especially, when N=3, μ=α=2, V=1 and g(x,u)=u2, equation (1.1) becomes the Choquard–Pekar equation

(1.2)-Δu+u=(I2|u|2)uinR3.

Pekar firstly introduced equation (1.2) in the quantum theory to characterize a polaron at rest in [31]. After two decades, Choquard [22] considered equation (1.2) as a certain approximation to Hartree–Fock theory of one component plasma to describe an electron trapped in its own hole. Afterwards, in [32], Penrose viewed the quantum state attenuation as a gravitational phenomenon and proposed equation (1.2) again to study the self-gravitational collapse of a quantum mechanical wave function.

Mathematically, the existence, multiplicity and qualitative properties of nontrivial solutions for equation (1.1) were widely studied in the last decades; see [1, 2, 3, 8, 10, 9, 17, 16, 13, 14, 22, 24, 25, 18, 21, 20, 19, 30, 28, 27, 29, 26, 35, 37, 36, 39, 41, 42] and the references therein. In [22], Lieb considered the physical case of equation (1.1), namely equation (1.2); there he proved the existence and uniqueness (up to translations) of positive radial ground state solution by rearrangement techniques. Later, Lions further studied the same problem and proved the existence of infinitely many radial solutions via the variational methods in [24]. As for the generalized Choquard equation (1.1), Ma and Zhao [26] assumed V(x)const., G(x,u)=|u|p and obtained that every positive solution is radially symmetric and monotone decreasing about some point under some assumptions on N,α and 𝑝. In [27], Moroz and Van Schaftingen considered the case of equation (1.1) that V(x)1, μ=1p and G(x,u)=|u|p; they proved the existence, regularity, radial symmetry and decaying property of ground state solutions when p(2*α,2α*). In addition, by establishing a Nehari–Pohožaev type identity, they also showed the nonexistence of nontrivial solutions for equation (1.1) when p(2*α,2α*). As described in the literature, the endpoints 2*α=N+αN and 2α*=N+αN-2 are called lower and upper critical exponents in terms of the Hardy–Littlewood–Sobolev inequality, respectively. It is easy to verify that the critical terms RN(Iα|u|2*α)|u|2*αdx and RN(Iα|u|2α*)|u|2α*dx are invariant under the scaling actions σN2u(σ) and σN-22u(σ), respectively. Furthermore, these two scaling actions serving as group actions on H1(RN) are noncompact. As a consequence, similar to the Sobolev critical case, the critical exponents 2*α and 2α* may yield two types of loss of compactness from the perspective of variational methods. Fortunately, these two kinds of loss of compactness can be recovered to some extent by using the extremal functions of the Hardy–Littlewood–Sobolev inequality as in [7].

In recent years, an increasing number of scholars paid attention to critical Choquard equations and proved some interesting results in [2, 8, 9, 17, 13, 14, 21, 20, 19, 29, 35, 37, 36, 39, 41, 42]. We refer readers to [2, 9, 17, 13, 14, 21, 20, 36] for the upper critical cases and [8, 29, 39, 41, 42] for the lower critical cases. Also, the Choquard type equations with doubly critical exponents were considered in [19, 35, 37]. More precisely, Seok [35] studied the equation

(1.3)-Δu+u=(Iα(12α*|u|2α*+12*α|u|2*α))(|u|2α*-2u+|u|2*α-2u)inRN,

where N5 and α(0,N-4), by using the compactness lemma of Strauss; he proved the existence of nontrivial radial solution in Hr1(RN), which denotes the space of radial functions in H1(RN). Recently, Su [37] extended Seok’s result of [35] in the sense that the existence of nontrivial solution for equation (1.3) is shown in H1(RN) instead of its subspace Hr1(RN). In [19], based on the work [28], Li and Ma obtained a positive radial ground state solution for equation (1.3) via the perturbed method.

Particularly, when it comes to the case of Vconst., Gao and Yang studied equation (1.1) with the periodic potential 𝒱 changing sign in [13], where they assumed that 0 lies in a gap of the spectrum of -Δ+V(x) and proved the existence of nontrivial solution by using a generalized linking theorem. In [1, 10, 21], the authors studied equation (1.1) with 𝒱 vanishing at infinity. Concretely, Alves, Figueiredo and Yang [1] proved the existence of positive solutions by the penalization method; Chen and Yuan [10] obtained a ground state solution via a non-Nehari manifold method and a diagonal approach. Later, Li, Li and Tang [21] extended the result of [10] from the noncritical case to the upper critical case. Additionally, equation (1.1) with the steep potential V(x)=λV(x)+b was studied in [3, 16, 25, 20, 18, 36]. In detail, Alves, Nóbrega and Yang [3] assumed that N=3, α(0,3), μ=(pAα)-1, λ>0, b=1, G(x,u)=|u|p and VC(R3,R) satisfies

  1. there exists some M>0 such that |{xR3:V(x)M}|<+,

  2. V(x)0, Ω:=intV-1(0) is a nonempty bounded set with smooth boundary, and Ω¯=V-1(0),

  3. Ω has 𝑘 connected components, more precisely, Ω=j=1kΩj with dist(Ωi,Ωj)>0 for all ij.

By employing a modified deformation lemma instead of penalizing the nonlinearity, they obtained the existence, multiplicity and concentration of multi-bump solutions for 𝜆 large enough and p(2,3+α). Afterwards, Guo and Hu [16] generalized the results in [3] to the more general case of equation (1.1). Lü [25] studied the case that N3, α(0,N), b=μ=1, G(x,u)=|u|p and VC(RN,R) satisfies
  1. there exists some M>0 such that |{xRN:V(x)M}|<+,

  2. V(x)0 in RN, Ω:=intV-1(0) is a nonempty set with smooth boundary and Ω¯=V-1(0).

By using the mountain pass theorem and the Nehari manifold methods, he proved the existence and concentration of ground state solutions to equation (1.1) for λ>0 large enough and any p(2*α,2α*). Later, Li et al. equipped the focusing equation of [25] with a Sobolev critical nonlinearity in R3 in [18], where they obtained the existence and concentration of ground state solutions for λ>0 large and 𝛼 small enough by employing the Nehari manifold methods. In [36], Shen, Gao and Yang supposed that N4, α(0,N), λ>0, μ=(2α*Aα)-1, G(x,u)=|u|2α*, moreover, VC(RN,R) may satisfy (V1), (V2) and
  1. lim inf|x|+V(x)>0.

Due to (V2), the operator -Δ has a sequence of Dirichlet eigenvalues 0<μ1<μ2μn+ in H01(Ω). If b(-μ1,0), (V1) and (V2) hold, they obtained the existence of ground state solution by the mountain pass theorem and proved the multiplicity of nontrivial solutions by using the Lusternik–Schnirelmann category theory for 𝜆 large enough. Moreover, if b<-μ1, b-μi for all iN+, (V2) and (V5) hold, by using a nonlinear superposition principle, they showed the existence of a ground state solution for 𝜆 large enough. Further, they proved the asymptotic behavior of ground state solutions as λ+. Recently, Li, Li and Tang [20] extended the results of ground state solution in [36] to the physical case N=3 of equation (1.1). For a complete discussion on equation (1.1), we refer readers to [30].

As we know, the steep well potential was introduced by Bartsch and Wang [5], where they studied

(1.4)-Δu+(λV(x)+b)u=h(x,u)inRN.

By assuming that λ>0, b=1, the function VC(RN,R) satisfies (V1) and the hypothesis

  1. V(x)0 in RN and Ω:=intV-1(0) is nonempty,

additionally, the function hC(RN×R,R) satisfies some mild subcritical conditions, they proved the existence and multiplicity of nontrivial solutions to equation (1.4) for large 𝜆. After this initial work, Bartsch and Wang [6] assumed h(x,u)=|u|p-2u, p(2,2NN-2), VC(RN,R) satisfies (V1) and (V4). By the Nehari manifold methods, they proved the existence and concentration of ground state solutions to equation (1.4) for λ>0 large enough. Meanwhile, by applying the Lusternik–Schnirelmann theory, they showed the multiplicity and concentration of positive solutions to equation (1.4) for λ>0 large enough and p<2* close to 2*, where 2*=2NN-2 is the Sobolev critical exponent. Later, the authors in [11, 40, 45] studied the critical cases of equation (1.4). In addition, as for the case of b=0, Ding and Szulkin showed the existence, multiplicity and concentration of nontrivial solutions to equation (1.4) for λ>0 large enough in [12], where the potential VC(RN,R) is allowed to be sign-changing.

Having recalled the above results, in the present paper, we study the Choquard equation

(1.5)-Δu+λV(x)u=(Iα(12α*|u|2α*+12*α|u|2*α))(|u|2α*-2u+|u|2*α-2u)inRN,

where N4, α(0,N), λ>0; additionally, the function 𝑉 satisfies (V1) and the following hypotheses:

  1. VC(RN,R) and 𝑉 is bounded from below;

  2. Ω:=intV-1(0) is a nonempty set with smooth boundary and Ω¯=V-1(0).

Now we are ready to present our main results. Firstly, we conclude the following result.

Theorem 1.1

Assume that N4, α(0,N), (V1)–(V3) are satisfied and V(x)0 in RN. Then there exists Λ>0 such that equation (1.5) possesses a positive ground state solution uλ for any λΛ. Further, for any sequence {λn}[Λ,+) satisfying λn+, the sequence of positive ground state solutions uλnu in H1(RN) in the sense of subsequence, where 𝑢 is a positive ground state solution of

(1.6){-Δu=Aα(Ω12α*|u(y)|2α*+12*α|u(y)|2*α|x-y|N-αdy)(|u|2α*-2u+|u|2*α-2u),xΩ,u(x)=0,xΩ.

Furthermore, if the function 𝑉 changes sign in RN, we will prove the existence of nontrivial weak solutions to equation (1.5) for λ>0 large enough. Namely, we deduce the second result of this paper.

Theorem 1.2

Assume that N>5, α(NN-4,N), (V1)–(V3) hold, VC0,r¯(RN) for some r¯(0,1) and V-0. Then there is Λ¯>0 such that equation (1.5) has a nontrivial solution uλ for any λΛ¯.

Remark 1.3

In contrast to [19, 35, 37], where the authors studied equation (1.5) with V(x)const., we generalize their results from the constant potential to the deepening well potential. As we shall see hereinafter, unlike the upper critical case, the loss of compactness in RN provoked by the lower critical exponent may be recovered by the steep potential λV(x) with 𝑉 satisfying (V1)–(V3). Comparing with [3, 16, 18, 20, 25, 36], in which the authors considered the Choquard equation that involves steep potential well and single critical (noncritical) exponent, we extend their results in the sense that the doubly critical case is studied in the present paper. Additionally, it is worth mentioning that the assumption N4 in Theorem 1.1 is merely used for recovering the lack of compactness provoked by the upper critical exponent to some extent, so does the assumption that N>5 and NN-4<α<N in Theorem 1.2.

To our knowledge, there seem to be no results on the Choquard equation (1.1) with steep potential of the form V(x)=λV(x), where 𝑉 satisfies (V1)–(V3) and is allowed to be sign-changing. Therefore, this paper can be regarded as a supplementary work on some related literature. We point out that our results here extend the ones in [12] from subcritical Schrödinger equation to doubly critical Choquard equation, and our extension is meaningful since the presence of nonlocal nonlinearity and doubly critical exponents would provoke some new difficulties. Regrettably, in Theorem 1.2, we say nothing about the asymptotic behavior of nontrivial solutions for equation (1.5) as λ+ since the boundedness of the sequence {uλnλn} cannot be derived for any given sequence {λn}[Λ¯,+) such that λn+.

Before proving Theorems 1.1 and 1.2, we set the variational frameworks for equations (1.5) and (1.6). Let E:={uH1(RN):RNV+(x)u2dx<+}, endowed with the inner product and the norm

(u,v)=RNuv+V+(x)uvdxandu=(u,u)12.

Clearly, 𝐸 is a Hilbert space. For λ>0, we also equip 𝐸 with the inner product and norm

(u,v)λ=RNuv+λV+(x)uvdxanduλ=(u,u)λ12.

It is easy to see that the norms and λ are equivalent for any fixed λ>0. Set Eλ=(E,λ). From (V1), the Hölder inequality and the continuous embedding D1,2(RN)L2*(RN), we conclude that 𝐸 is continuously embedded in H1(RN). Thereby, for any s[2,2*], 𝐸 is continuously embedded in Ls(RN), and there exists some constant νs>0 such that |u|sνsu for all uE.

The weak solution of equation (1.5) can be caught as the critical point of the energy functional

Iλ(u)=12RN|u|2+λV(x)u2dx-12RN(Iα(12α*|u|2α*+12*α|u|2*α))(12α*|u|2α*+12*α|u|2*α)dx.

Similarly, the weak solution of equation (1.6) corresponds to the critical point of the energy functional

I(u)=12Ω|u|2dx-Aα2ΩΩ(12α*|u(x)|2α*+12*α|u(x)|2*α)(12α*|u(y)|2α*+12*α|u(y)|2*α)|x-y|N-αdxdy.

By using the Hardy–Littlewood–Sobolev (H-L-S for short) inequality [23, Theorem 4.3] and the Sobolev inequality, we standardly deduce that IλC1(Eλ,R) for any λ>0 and IC1(H01(Ω),R).

To recover the loss of compactness provoked by the H-L-S critical exponents, we introduce

(1.7)S¯α=inf{RN|u|2dx:uL2(RN)andRN(Iα|u|2*α)|u|2*αdx=1},
(1.8)Sα=inf{RN|u|2dx:uD1,2(RN)andRN(Iα|u|2α*)|u|2α*dx=1}.
As we shall see thereinafter, unlike the upper critical case, the loss of compactness provoked by lower critical exponent may be recovered by the appearance of steep potential well. For the upper critical case, Gao and Yang [13, Lemma 2.6] verified that Sα is achieved by 𝑢 if and only if, for any xRN,

u(x)=(const.)Ub,x0(x)=(const.)[N(N-2)b2]N-24[C(N,α)AαSα2]N-24+2α(b2+|x-x0|2)N-22

for some b(0,+) and x0RN, where

C(N,α):=πN-α2Γ(α2)Γ(N+α2)[Γ(N)Γ(N2)]αN

is the sharp constant in the H-L-S inequality and 𝑆 is the best Sobolev constant. Moreover, the minimizer U1,0 of Sα satisfies

(1.9)RN|U1,0|2dx=RN(Iα|U1,0|2α*)|U1,0|2α*dx=SαN+α2+α.

In the forthcoming sections, we will present the proof of Theorem 1.1 in Section 2 and the proof of Theorem 1.2 in Section 3. To facilitate reading, throughout this paper, we use the following notations.

  • |β|=i=1Nβi for β=(β1,,βN)NN and

    Dβu=|β|ux1β1xNβN.

    For kN and r[0,1],

    Ck(RN):={u:RNR:Dβuis continuous inRNfor anyβNNwith|β|k},
    Ck,r(RN):={uCk(RN):supxyRN|Dβu(x)-Dβu(y)||x-y|r<for anyβNNwith|β|k}.

  • C0(RN) consists of infinitely times differentiable functions with compact support in RN.

  • Lp(RN) is the usual Lebesgue space with the norm |u|p=(RN|u|pdx)1p for p[1,+).

  • L(RN) is the space of measurable functions with the norm |u|=esssupxRN|u(x)|.

  • H1(RN){uL2(RN):|u|L2(RN)} endowed with the inner product and the norm

    (u,v)H=RNuv+uvdxanduH=(u,u)H12.
  • Lp(Ω) is the usual Lebesgue space with the norm |u|p,Ω=(Ω|u|pdx)1p for p[1,+).

  • D1,2(RN) is the completion of C0(RN) with respect to the norm uD=(RN|u|2dx)12.

  • The best Sobolev constant S:=inf{uD2:uD1,2(RN)and|u|2*=1}.

  • V±(x):=max{±V(x),0} and (E*,*) is the dual space of Banach space (E,).

  • o(1) is a quantity tending to 0 as n and |Ω| is the Lebesgue measure of ΩRN.

  • For r>0 and yRN, Br(y){xRN:|x-y|<r}, Brc(y)RN\Br(y) and Br(0)Br.

2 Proof of Theorem 1.1

It is clear that V+=V in the case of V(x)0 in RN. To study the existence of a ground state solution for equation (1.5), we define the corresponding Nehari manifold and the least energy as follows:

Nλ={uEλ\{0}:Iλ(u),u=0}andmλ=inf{Iλ(u):uNλ}.

Similarly, for the limit problem of equation (1.5), namely equation (1.6), we define the Nehari manifold

N={uH01(Ω)\{0}:I(u),u=0}andm=inf{I(u):uN}.

Since Nλ is not of class C1, the ground state solution of equation (1.5) cannot be obtained by directly using the minimization arguments on Nλ. Following [44], by defining a homeomorphism between the unit sphere of Eλ and Nλ, we will show that mλ equals the mountain pass value of Iλ for any λ>0. Thereby, we shall obtain the ground state solutions of equation (1.5) via the mountain pass theorem.

Lemma 2.1

For any λ>0, the functional Iλ possesses a mountain pass geometry in Eλ. Namely,

  1. there exist constants ρλ,σλ>0 such that Iλ|Sρλσλ, where Sρλ:={uEλ:uλ=ρλ};

  2. there exists a function ωEλ such that ωλ>ρλ and Iλ(ω)<0.

Proof

(1) By using the H-L-S, Minkowski and Sobolev inequalities, we conclude that, for any uEλ,

Iλ(u)12uλ2-12C(N,α)Aα(RN||u|2α*+|u|2*α|2NN+α)N+αN12uλ2-C(N,α)Aα[(ν2u)22*α+(ν2*u)22α*].

Thereby, we may choose sufficiently small ρλ>0 and σλ>0 such that conclusion (1) is satisfied.

(2) It is obvious that limt+Iλ(tu)=- for any uEλ\{0}. Thus this lemma is proved. ∎

Recall that {un}Eλ is called a (PS)c sequence of Iλ at cR if Iλ(un)nc and Iλ(un)n0 in Eλ*. Due to Lemma 2.1, by the mountain pass theorem [44, Theorem 2.10], Iλ has a (PS)cλ sequence for any λ>0. For any λ>0 and each uEλ\{0}, it is easy to show that there exists a unique tu>0 such that Iλ(tuu)=maxt0Iλ(tu) and tuuNλ. Thereby, we can verify as in [44, Theorem 4.2] that

(2.1)mλ=cλ:=infγΓλsupt[0,1]Iλ(γ(t))

for any λ>0, where the family of continuous paths Γλ={γC([0,1],Eλ):γ(0)=0,Iλ(γ(1))<0}.

Due to (V3), without losing generality, we may assume 0Ω and B2δ¯Ω for some δ>0. Take a cut-off function φC0(RN,[0,1]) such that φ(x)1 in Bδ, while φ(x)0 in B2δc, and set vε=φUε,0 for ε>0. Denote the restriction of vε to Ω as uε; since Ω is smooth, then uεH01(Ω) for any ε>0. Further, computing as in [7, 9, 13, 44], by (1.9) and the invariance of dilation, we conclude, as ε0+,

(2.2)Ω|uε|2dx=SαN+α2+α+O(εN-2),
(2.3)ΩΩ|uε(x)|2α*|uε(y)|2α*|x-y|N-αdxdy=Aα-1SαN+α2+α+O(εN+α2).

Lemma 2.2

For any λ>0, the least energy mλ(0,m] and

m<m*2+α2(N+α)(N+αN-2)N-22+αSαN+α2+α.

Proof

It follows from (2.1) and Lemma 2.1 (1) that mλ>0. Since NNλ and Iλ=I on N, we deduce mλm. Then it suffices to show m<m*. For any ε>0, since I(tuε)>0 for t>0 small enough and limt+I(tuε)-, there exists some tε>0 such that I(tεuε)=maxt0I(tuε), I(tεuε),tεuε=0. Noting that I(tεuε)m>0 for any ε>0, we conclude that there exist some ε1>0 and T1,T2>0 such that tε[T1,T2] for all ε(0,ε1). For t[0,+) and ε>0, define

Fε(t)=t22Ω|uε|2dx-Aαt22α*2(2α*)2ΩΩ|uε(x)|2α*|uε(y)|2α*|x-y|N-αdxdy.

Clearly, Fε has a maximum on [0,+) for any ε>0. By (2.2) and (2.3), we obtain that, as ε0+,

(2.4)maxt0Fε(t)=2+α2(N+α)(N+αN-2)N-22+α(Ω|uε|2dx)N+α2+α(AαΩΩ|uε(x)|2α*|uε(y)|2α*|x-y|N-αdxdy)N-22+α=2+α2(N+α)(N+αN-2)N-22+αSαN+α2+α+O(εN-2)-O(εN+α2).

Now, noting min{N-2,N+α2}>2*α since N4, by (2.2)–(2.4), we have, for ε(0,δ) small enough,

I(tεuε)Fε(tε)-Aαtε2α*+2*α2α*2*αBεBε|uε(x)|2α*|uε(y)|2*α|x-y|N-αdxdymaxt0Fε(t)-AαT12α*+2*αε2*α2α*2*αB1B1|U1,0(x)|2α*|U1,0(y)|2*α|x-y|N-αdxdy<2+α2(N+α)(N+αN-2)N-22+αSαN+α2+α.

Then, by the fact that I(tεuε)m for all ε>0, we have m<m*. Thus this lemma is proved. ∎

In the following lemma, we give some properties of the (PS)c sequence for the functional Iλ.

Lemma 2.3

There exists Λ>0 such that, for any λΛ and (PS)c sequence {un} of the functional Iλ at level c(0,m*), there exist a bounded sequence {zn}RN and some constant η¯>0 satisfying

(2.5)B1(zn)|un|2dxη¯.

Proof

Take Λ1[(MS¯α)-1(NN+α)N-αN+α(2Nm*α)αN+α,+). It follows from the definition of {un} that

(2.6)m*+o(1)+o(unλ)Iλ(un)-122*αIλ(un),unα2(N+α)unλ2.

Then there exists C1>0, independent of 𝜆, such that lim supnunλC1 for any λ[Λ1,+). It is obvious that {un} is bounded in Eλ. Below, we end the proof in two steps. Firstly, we show that, for any λΛ1 and (PS)c sequence {un} of Iλ with c(0,m*), there exists a sequence {zn}RN such that (2.5) holds. If not, the Lions lemma [44, Lemma 1.21] implies that un0 in Lp(RN) for any p(2,2*). Thereby, from (V1), (V2), the boundedness of {unλ}, the Hölder and H-L-S inequalities,

(2.7)limn{xRN:V(x)M}un2dx=0,
(2.8)limn{xRN:V(x)M}V(x)un2dx=0,
(2.9)limnRN(Iα|un|2α*)|un|2*αdx=0.
With the fact Iλ(un),un=o(1) in hand, we then deduce from (2.8) and (2.9) that, for any λΛ1,

(2.10)RN|un|2dx+λ{xRN:V(x)>M}V(x)un2dx=12α*RN(Iα|un|2α*)|un|2α*dx+12*αRN(Iα|un|2*α)|un|2*αdx+o(1).

Additionally, for any given λΛ1, it follows from the fact Iλ(un)nc and (2.8)–(2.10) that

(2.11)c+o(1)=2α*-12(2α*)2RN(Iα|un|2α*)|un|2α*dx+2*α-12(2*α)2RN(Iα|un|2*α)|un|2*αdx.

Further, from (2.10), (2.7), the definitions of S¯α,Sα in (1.7) and (1.8), we deduce that, for any λΛ1,

(2.12)Sα(RN(Iα|un|2α*)|un|2α*dx)12α*+Λ1MS¯α(RN(Iα|un|2*α)|un|2*αdx)12*αRN|un|2dx+Λ1M{xRN:V(x)>M}un2dx+o(1)12α*RN(Iα|un|2α*)|un|2α*dx+12*αRN(Iα|un|2*α)|un|2*αdx+o(1).

Denote lim supnRN(Iα|un|2α*)|un|2α*dx=l1 and lim supnRN(Iα|un|2*α)|un|2*αdx=l2. It is clear that l1,l2[0,+). Passing to the limit in (2.12) as n, we conclude the four possibilities:

(1)l1=l2=0;
(2)l1=0andl2(Λ12*αMS¯α)2*α2*α-1;
(3)l1(2α*Sα)2α*2α*-1andl2=0;
(4)l1(2α*Sα)2α*2α*-1orl2(Λ12*αMS¯α)2*α2*α-1ifl1l20.
In case (1), by (2.11), we have c=0, which contradicts the premise c>0. Similarly, in every case of (2)–(4), from (2.11) again, we deduce cm*, which is impossible since c<m*. Therefore, for any λΛ1 and (PS)c sequence {un} of Iλ with c(0,m*), there is {zn}RN such that (2.5) holds.

Next, we prove that such {zn} is bounded in RN for λΛ1 large enough. By (V1), there exists R>0 such that |{xBRc:V(x)M}|(η¯S4C12)N2. Then, from the Hölder and Sobolev inequalities,

(2.13)lim supn{xBRc:V(x)M}un2dx|{xBRc:V(x)M}|2NS-1lim supnunλ2η¯4.

Additionally, choosing Λ=max{Λ1,4C12η¯M}, we obtain by the fact V(x)0 in RN that, for all λΛ,

(2.14)lim supn{xBRc:V(x)>M}un2dx1λMlim supnunλ2C12λMη¯4.

Now, suppose inversely |zn| up to a subsequence for some λΛ, (2.5), (2.13) and (2.14) imply

η¯lim supnB1(zn)un2dxlim supn({xBRc:V(x)M}+{xBRc:V(x)>M})un2dxη¯2,

which is a contradiction. Thus such {zn}RN is bounded for any λΛ, and this lemma is proved. ∎

Proof of Theorem 1.1

Firstly, we show the existence of a positive ground state solution to equation (1.5) for any fixed λΛ. With Lemmas 2.1 and 2.2 in hand, we deduce from (2.1) and the mountain pass theorem [44, Theorem 2.10] that Iλ possesses a (PS)mλ sequence {vn}Eλ with mλ(0,m*). Repeating the estimates in (2.6), we deduce that {vn} is bounded in Eλ. Thereby, according to the Rellich–Kondrakov theorem, there exists some vEλ such that vnv in Eλ, vnv in Llocs(RN) for any s[1,2*) and vn(x)v(x) a.e. in RN in the sense of subsequence. Using a standard argument, we obtain that Iλ(v)=0 in Eλ*. It follows from Lemma 2.3 that v0. Then, from the weakly lower semicontinuity of the norm λ, the Fatou lemma and the boundedness of {vn} in Eλ, we conclude

mλIλ(v)-122*αIλ(v),v=α2(N+α)vλ2+1N+αRN12α*(Iα|v|2α*)|v|2α*+12*α(Iα|v|2α*)|v|2*αdxlim infn[α2(N+α)vnλ2+1N+αRN12α*(Iα|vn|2α*)|vn|2α*+12*α(Iα|vn|2α*)|vn|2*αdx]=limn[Iλ(vn)-122*αIλ(vn),vn]=mλ.

That is, Iλ(v)=mλ. Further, since |v|Nλ and Iλ(|v|)=mλ, we can show as in [21] that Iλ(|v|)=0. Consequently, we may assume that v(x)0 in RN. Then, by the strong maximum principle of weak solution [15, Theorem 8.19], we obtain v(x)>0 in RN. From the above discussion, equation (1.5) has a positive ground state solution for any λΛ; we mark it as uλ for convenience of arguments below.

Next, we investigate the asymptotic behavior of the positive ground state solutions for equation (1.5) as λ+. For every sequence {λn}[Λ,+) satisfying λn+, let uλnNλn such that uλn(x)>0 in RN, Iλn(uλn)=0 in Eλn* and Iλn(uλn)=mλn for every 𝑛. From Lemma 2.2, it follows that {mλn}(0,m]. It is easy to verify that m<+. Furthermore, since (2.1) implies that mλ is nondecreasing with respect to 𝜆 in [Λ,+), there results {mλn}[mΛ,m]. Then we may assume Iλn(uλn)m(0,m] in the sense of subsequence. From the definition of {uλn}, we conclude that

(2.15)mIλn(uλn)-122*αIλn(uλn),uλnα2(N+α)uλnλn2.

It is obvious that {uλnΛ} is bounded. Moreover, by the continuous embedding EΛH1(RN), we know that {uλn} is bounded in H1(RN). Then there exists uH1(RN) such that, up to subsequences,

(2.16){uλnuinH1(RN),uλnuinLlocs(RN)for anys[1,2*),uλn(x)u(x)a.e. inRN.

We claim that u(x)0 on Ωc:=RN\Ω. If not, there exists some compact set ΩΩc such that |Ω|>0, dist(Ω,Ω)>0 and u|Ω0. From minxΩV(x)>0, the definition of {uλn} and (2.16), we have

mλn=Iλn(uλn)-122*αIλn(uλn),uλnα2(N+α)RN|uλn|2+λnV(x)uλn2dxλnα2(N+α)ΩminxΩV(x)uλn2dx+asn,

which contradicts limnmλn=m<+, i.e., u(x)0 on Ωc. Since Ω is smooth, there results uH01(Ω). Recalling Iλn(uλn)=0 in Eλn* and V|Ω=0, we conclude that, for any ψH01(Ω),

RNuλnψdx=RN(Iα(12α*|uλn|2α*+12*α|uλn|2*α))(|uλn|2α*-2uλn+|uλn|2*α-2uλn)ψdx.

Then, letting n, we deduce from the weakly sequential continuity of I0 that, for all ψH01(Ω),

Ωuψdx=AαΩΩ(12α*|u(y)|2α*+12*α|u(y)|2*α)(|u(x)|2α*-2u(x)+|u(x)|2*α-2u(x))ψ(x)|x-y|N-αdxdy.

As a consequence, 𝑢 is a weak solution of equation (1.6). In what follows, we shall further prove that uλnu in H1(RN) up to a subsequence and 𝑢 is a positive ground state solution of equation (1.6).

Set vλn=uλn-u for each 𝑛. Due to (2.15) and u(x)0 on Ωc, there exists C2>0 such that vλnλnC2 for all 𝑛. By (2.16), we have vλn20 in LNN-2(RN). Thereby, it follows from (V1) that

(2.17)RNvλn2dx={xRN:V(x)>M}vλn2dx+{xRN:V(x)M}vλn2dx1λnMRNλnV(x)vλn2dx+o(1)C22λnM+o(1)0asn.

We derive from the H-L-S inequality and (2.17) that

(2.18)limnRN(Iα|vλn|2*α)|vλn|2*αdxAαC(N,α)limn|vλn|222*α=0.

Since uλnu in L2(RN), up to a subsequence, there is a function ωL2(RN) such that, for all 𝑛,

||uλn(x)|2*α-|u(x)|2*α|2NN+α2N-αN+α(|uλn(x)|2+|u(x)|2)2(|ω(x)|2+|u(x)|2)a.e. inRN.

Then, noting that (2.16) implies ||uλn(x)|2*α-|u(x)|2*α|2NN+α0 a.e. in RN, we derive from the H-L-S inequality, the boundedness of {|uλn|2*} and the dominated convergence theorem that

(2.19)|RN(Iα|uλn|2α*)(|uλn|2*α-|u|2*α)dx|AαC(N,α)|uλn|2*2α*(RN||uλn|2*α-|u|2*α|2NN+αdx)N+α2N0asn.

Because {uλn} is bounded in L2*(RN) and uλn(x)u(x) a.e. in RN, |uλn|2α*|u|2α* in L2NN+α(RN). Recall that the Riesz potential defines a bounded linear map from L2NN+α(RN) to L2NN-α(RN), we have

Iα|uλn|2α*Iα|u|2α*inL2NN-α(RN).

Thereby, it follows from |u|2*αL2NN+α(RN) that

RN(Iα|uλn|2α*-Iα|u|2α*)|u|2*αdx0asn,

which together with (2.19) implies

(2.20)limnRN(Iα|uλn|2α*)|uλn|2*αdx=RN(Iα|u|2α*)|u|2*αdx.

By (2.16), the nonlocal Brézis–Lieb lemma [9, Lemma 2.2], (2.18), (2.20) and I(u),u=0, we obtain

(2.21)0=Iλn(uλn),uλn=RN|vλn|2+|u|2+λnV(x)vλn2dx-12α*RN(Iα|vλn|2α*)|vλn|2α*+(Iα|u|2α*)|u|2α*dx-12*αRN(Iα|vλn|2*α)|vλn|2*α+(Iα|u|2*α)|u|2*αdx-(12α*+12*α)RN(Iα|uλn|2α*)|uλn|2*α+o(1)=I(u),u+vλnλn2-12α*RN(Iα|vλn|2α*)|vλn|2α*dx+o(1)=vλnλn2-12α*RN(Iα|vλn|2α*)|vλn|2α*dx+o(1).

Setting κ=lim supnvλnλn2, we deduce from (2.21), (1.8) and V(x)0 in RN that 2α*Sα2α*κκ2α*.

Accordingly, there results

κ=0orκ(N+αN-2)N-22+αSαN+α2+α.

In the latter case, from (2.16), the nonlocal version of Brézis–Lieb lemma [9, Lemma 2.2], (2.18), (2.20), (2.21) and I(u),u=0, we obtain that

m=limnIλn(uλn)=limn[12RN|vλn|2+|u|2+λnV(x)vλn2dx-12(2α*)2RN(Iα|vλn|2α*)|vλn|2α*+(Iα|u|2α*)|u|2α*dx-12(2*α)2RN(Iα|u|2*α)|u|2*αdx-12α*2*αRN(Iα|u|2α*)|u|2*αdx]=limn[I(u)+12vλnλn2-12(2α*)2RN(Iα|vλn|2α*)|vλn|2α*dx]=I(u)-122*αI(u),u+2+α2(N+α)lim supnvλnλn22+α2(N+α)(N+αN-2)N-22+αSαN+α2+α,

which is impossible since m<m*. Hence, limnvλnλn=0. It is clear that vλnD0. Then it follows from (2.17) that uλnu in H1(RN) in the sense of subsequence. It is easy to see that u0. Moreover, by (2.16), the weakly lower semicontinuity of the norm D and the Fatou lemma, we have

mI(u)=I(u)-122*αI(u),u=α2(N+α)Ω|u|2dx+AαN+αΩΩ12α*|u(x)|2α*|u(y)|2α*+12*α|u(x)|2α*|u(y)|2*α|x-y|N-αdxdy12(N+α)lim supn[αuλnλn2+2RN(Iα(12α*|uλn|2α*+12*α|uλn|2*α))|uλn|2α*dx]=limn[Iλn(uλn)-122*αIλn(uλn),uλn]=mm.

Namely, I(u)=m. It is clear that u(x)0 a.e. in Ω. Then the strong maximum principle implies u(x)>0 in Ω. Thus, 𝑢 is a positive ground state solution of equation (1.6), and Theorem 1.1 is proved. ∎

3 Proof of Theorem 1.2

In this section, we intend to complete the proof of Theorem 1.2 by establishing some preliminary lemmas. To begin with, we investigate some properties on the Hilbert space Eλ when V-0. Define

F={uE:sptuV-1([0,+))}.

Let Fλ be the orthogonal complement of 𝐹 in Eλ. We define Lλ=-Δ+λV(x) and the bilinear form

aλ(u,v)=RNuv+λV(x)uvdx.

It is easy to verify that the operator Lλ is formally self-adjoint in L2(RN) and aλ is continuous in Eλ. Motivated by [12], for any given λ>0, we outline some results of the eigenvalue problem

(3.1)-Δu+λV+(x)u=μλV-(x)u,uFλ.

It follows from (V1), (V2) that the functional Φ(u)=RNV-(x)u2dx is weakly continuous in Fλ. Then, by [43, Theorems 4.45 and 4.46], (3.1) has a sequence of positive eigenvalues {μj(λ)} characterized by

μj(λ)=infMFλ,dimMjsup{uλ2:uMandRNλV-(x)u2dx=1}for anyjN+,

where μ1(λ)μ2(λ)μj(λ)j+. Take eigenfunction ej(λ) corresponding to μj(λ) for each jN+ such that {ej(λ)} serves as an orthonormal basis of Fλ. For any λ>0, define the subspaces

Eλ-=span{ej(λ):μj(λ)1}andEλ+=span{ej(λ):μj(λ)>1}.

It is clear that Eλ=Eλ-Eλ+F and dimEλ-<+ for any λ>0. Fixed jN+, since [12, Lemma 2.1 (i)] implies limλ+μj(λ)=0, there exists some Λ¯1>0 such that dimEλ-1 for any λΛ¯1. Furthermore, it is easy to certify that the quadratic form aλ is negative semidefinite on Eλ-, positive definite on Eλ+F and aλ(u,v)=0 once u,v are from different subspaces of the decomposition of Eλ.

In order to prove Theorem 1.2, we need the following L and regular estimates of {ej(λ)}.

Lemma 3.1

Under the assumptions of Theorem 1.2, ej(λ)L(RN)C2(RN) for any 𝑗 and λ>0.

Proof

More generally, fixing λ>0 and μ>0, we shall verify that each weak solution 𝑢 of equation (3.1) satisfies uL(RN)C2(RN). Firstly, similar to the argument of [4, Proposition 2.1], by using the Moser iteration, we certify the L estimate on 𝑢. For each kN+ and any τ>1, define the sets

Ak,τ={xRN:|u(x)|τ-1k}andBk,τ=RN\Ak,τ.

Moreover, we introduce the functions

uk,τ(x)={|u(x)|2(τ-1)u(x),xAk,τ,k2u(x),xBk,τ,  and  wk,τ(x)={|u(x)|τ-1u(x),xAk,τ,ku(x),xBk,τ.

Clearly, uk,τ,wk,τEλ and uk,τ|u|2τ-1, wk,τ2=uuk,τ|u|2τ in RN. By direct calculation, we have

RNuuk,τdx=(2τ-1)Ak,τ|u|2(τ-1)|u|2dx+k2Bk,τ|u|2dx

and then

RN|wk,τ|2+λV+(x)wk,τ2dx-RNuuk,τ+λV+(x)uuk,τdx=(τ-1)2Ak,τ|u|2(τ-1)|u|2dx.

As a consequence, recalling 𝑢 weakly solves equation (3.1) and taking uk,τ as a test function, we deduce

RN|wk,τ|2+λV+(x)wk,τ2dx[(τ-1)22τ-1+1]RNuuk,τ+λV+(x)uuk,τdx=τ22τ-1RNμλV-(x)uuk,τdxτ2μλ|V-|RNwk,τ2dx,

where we used the fact that 𝑉 is bounded from below. Thereby, it follows from the definition of 𝑆 that

(Ak,τ|wk,τ|2*dx)22*(RN|wk,τ|2*dx)22*τ2μλ|V-|S-1RNwk,τ2dx.

In view of this fact, observing that |wk,τ|=|u|τ in Ak,τ and |wk,τ||u|τ in RN, we conclude that

(3.2)(Ak,τ|u|τ2*dx)22*τ2μλ|V-|S-1RN|u|2τdx.

Further, once uL2τ(RN), by the monotone convergence theorem, letting k in (3.2), we obtain

(3.3)|u|τ2*τ1τ(μλ|V-|S-1)12τ|u|2τ.

Now, setting σ=2*2 and taking τ=σ,σ2,,σn, nN+, in (3.3) in turn, by iterating, we conclude

|u|σn2*σj=1njσj(μλ|V-|S-1)12j=1n1σj|u|2*.

Whence, letting n, since the series j=1jσj=14N(N-2) and j=11σj=12(N-2), there results

|u|=limn|u|σn2*(NN-2)14N(N-2)(μλ|V-|S-1)14(N-2)|u|2*.

Secondly, we turn to studying the regular estimates on 𝑢. Since λ|μV--V+|λ(V++μV-) in RN and VC(RN,R) implies λ(V++μV-)LlocN2(RN), it follows from the Brézis–Kato estimate that uLlocp(RN) for any p[1,+). Then, by applying the Calderón–Zygmund estimate, we deduce that uWloc2,p(RN) for any p[1,+). Thereby, as a consequence of the general Sobolev embedding, there results that uCloc1,r(RN) for any r(0,1). Further, noting VC0,r¯(RN) for some r¯(0,1), we conclude from [15, Theorem 9.19] that uC2,r¯(RN) for r¯(0,1). Thus this lemma is proved. ∎

Let λΛ¯1 below. For any eigenfunction e(λ) corresponding to the first eigenvalue μ1(λ), thanks to [43, Lemma 4.43], we may assume e(λ)0 in RN. Thereby, from the strong maximum principle of weak solutions (see e.g. [15, Theorem 8.19]), it follows that e(λ)>0 in RN. Based on this fact, for any given λΛ¯1, by using the similar argument of [38, Lemma 3.3], we conclude the following result.

Lemma 3.2

Let vEλ- be such that v=0 on the set Ω0:={xRN:V(x)<0}. Then v=0 in RN.

Proof

Denote all distinct eigenvalues of (3.1) as μ1λ<<μnλλ1<μnλ+1λ<<μjλj+. Suppose inversely v0 in RN. We may assume v=v1+v2++vnλ, where vj is an eigenfunction corresponding to μjλ for any j{1,2,,nλ}. Due to v=0 on Ω0, we deduce from Lemma 3.1 that

0=μ1λλV-(x)v+Δv-λV+(x)v=λV-(x)[(μ1λ-μ2λ)v2++(μ1λ-μnλλ)vnλ]onΩ0.

Then we may replace vnλ by 0 in the decomposition of 𝑣 restricted on Ω0. Repeating this process (nλ-1) times, we conclude that 𝑣 equals some eigenfunction corresponding to μ1λ on Ω0, which is impossible since each eigenfunction corresponding to μ1λ is positive in RN. Thus this lemma is proved. ∎

Recall that a sequence {un}Eλ is called a (Ce)c sequence for the functional Iλ if

Iλ(un)ncandIλ(un)(1+unλ)n0.

In the forthcoming lemma, we will verify that the functional Iλ possesses a (Ce)c sequence in Eλ by applying the well-known linking theorem [34, Theorem 5.3] (see also [33, 44]).

From now on, we appoint N>5 and α(NN-4,N) (that are not optimal range for estimates (3.4)–(3.7)). Following the discussion before Lemma 2.2, by direct computation, we know, as ε0+,

(3.4)Ω|uε|2dx=O(ε2),
(3.5)Ω|uε|2αN+αdx=O(ε(N-2)αN+α),
(3.6)Ω|uε|2*dx=[AαC(N,α)]-NN+αSαN2+α+O(εN)
and

(3.7)β(ε)Ω|uε|2*(2+α)N+αdx={O(εN(2+α)N+α),N>4+α,O(εN2|lnε|),N=4+α,O(εN(N-2)N+α),N<4+α.
Lemma 3.3

For any λΛ¯1, the functional Iλ has the linking structure of [34, Theorem 5.3]. Namely,

  1. there exist rλ>0 and ηλ>0 such that Iλ(w)ηλ for all w{ωEλ+F:ωλ=rλ};

  2. for any fixed ε>0, there exists some large Rλ>rλ such that supwQλIλ(w)0, where

    Qλ:={w=v+tuε:vEλ-,t0,wλRλ}.

Proof

(1) By μ1(λ)μ2(λ)μj(λ)j+ and the definition of Eλ+, there exists some δλ>0 such that aλ(u,u)δλuλ2 for any uEλ+. Obviously, aλ(v,v)=vλ2 for all vF. Then, for any wEλ+F, setting w=u+v with uEλ+ and vF, by the H-L-S, Minkowski and Sobolev inequalities,

Iλ(w)12δλuλ2+12vλ2-12C(N,α)Aα(RN||w|2α*+|w|2*α|2NN+α)N+αN12min{δλ,1}wλ2-C(N,α)Aα[(ν2w)2(N+α)N+(ν2*w)2(N+α)N-2].

As a consequence, we may choose rλ>0 and ηλ>0 small enough such that conclusion (1) holds.

(2) Repeating the arguments in [14, Lemma 2.3], we conclude that the functionals

1:=(RN(Iα||2α*)||2α*dx)122α*,2:=(RN(Iα||2*α)||2*αdx)122*α

define two norms in Eλ. Noting that the bilinear form aλ is negative semidefinite on Eλ-, aλ(v,v)=vλ2 for any vF, and all norms of Eλ-Ruε are equivalent since dim(Eλ-Ruε)<+, we conclude

Iλ(w)12wλ2-12(2*α)2w222*α-12(2α*)2w122α*-

for wEλ-Ruε with wλ. Then there exists Rλ>rλ such that Iλ(w)0 for all wEλ-Ruε with wλ=Rλ. Further, Iλ(w)0 for all wEλ- with wλRλ. Thus this lemma is proved. ∎

In the following lemma, we will make an estimate on supwQλIλ(w) for ε>0 small enough and all λΛ¯1. For any λΛ¯1, since dimEλ-<+, Lemma 3.1 implies that vL(RN) for all vEλ-.

Lemma 3.4

For any λΛ¯1, there is ε¯>0 such that, for any ε(0,ε¯), the functional Iλ satisfies

(3.8)supvEλ-,t0Iλ(v+tuε)<m*2+α2(N+α)(N+αN-2)N-22+αSαN+α2+α.

Proof

Set Ωφ:=sptφ with 𝜑 introduced before Lemma 2.2. On the basis of Lemma 3.2, by applying a similar discussion to the verification of [14, Lemma 2.3], we conclude that the functionals

NL1:=(RN\ΩφRN\Ωφ||2α*||2α*|x-y|N-αdxdy)122α*,NL2:=(RN\ΩφRN\Ωφ||2*α||2*α|x-y|N-αdxdy)122*α

define two norms on Eλ-. It is easy to show that |a+b|pap+pap-1b for any a[0,+), bR and p>1. By the semigroup property of the Riesz potential that Iα=Iα2Iα2 for all α(0,N), we know

RN(Iα(|uε|2α*-1v))|uε|2α*-1vdx=RN(Iα2(|uε|2α*-1v))2dx0.

Then it follows from the Fubini theorem, the H-L-S inequality, (3.6) and (3.7) that, as ε0+,

(3.9)RNRN|v(x)+tuε(x)|2α*|v(y)+tuε(y)|2α*|x-y|N-αdxdyΩφΩφ|v(x)+tuε(x)|2α*|v(y)+tuε(y)|2α*|x-y|N-αdxdy+vNL122α*t22α*ΩΩ|uε(x)|2α*|uε(y)|2α*|x-y|N-αdxdy+vNL122α*+22α*t22α*-1ΩΩ|uε(x)|2α*|uε(y)|2α*-1v(y)|x-y|N-αdxdyt22α*ΩΩ|uε(x)|2α*|uε(y)|2α*|x-y|N-αdxdy+vNL122α*-22α*C(N,α)t22α*-1|uε|2*,Ω2α*|v|(Ω|uε|2*(2+α)N+αdx)N+α2Nt22α*ΩΩ|uε(x)|2α*|uε(y)|2α*|x-y|N-αdxdy+vNL122α*-42α*C(N,α)12SαN+α2(2+α)t22α*-1|β(ε)|N+α2N|v|.

Similarly, we deduce from Iα=Iα2Iα2, the Fubini theorem, the H-L-S inequality, (3.4) and (3.5) that

(3.10)RNRN|v(x)+tuε(x)|2*α|v(y)+tuε(y)|2*α|x-y|N-αdxdyt22*αΩΩ|uε(x)|2*α|uε(y)|2*α|x-y|N-αdxdy+vNL222*α-22*αC(N,α)t22*α-1O(ε2+α2)|v|asε0+.

Also, by the semigroup property of Riesz potential, the H-L-S inequality and (3.5)–(3.7), as ε0+,

(3.11)RNRN|v(x)+tuε(x)|2α*|v(y)+tuε(y)|2*α|x-y|N-αdxdyΩΩ|tuε(x)|2α*[|tuε(y)|2*α-2*α|tuε(y)|2*α-1|v(y)|]|x-y|N-αdxdy-2α*ΩΩ|tuε(x)|2α*-1|v(x)||v(y)+tuε(y)|2*α|x-y|N-αdxdyt2α*+2*αΩΩ|uε(x)|2α*|uε(y)|2*α|x-y|N-αdxdy-2*αC(N,α)t2α*+2*α-1|uε|2*,Ω2α*|v|(Ω|uε|2αN+αdx)N+α2N-22α*C(N,α)t2α*-1|v|2,Ω2*α|v|(Ω|uε|2*(2+α)N+αdx)N+α2N-22α*C(N,α)t2α*+2*α-1|uε|2,Ω2*α|v|(Ω|uε|2*(2+α)N+αdx)N+α2Nt2α*+2*αΩΩ|uε(x)|2α*|uε(y)|2*α|x-y|N-αdxdy-22α*C(N,α)t2α*-1|β(ε)|N+α2N|v|22*α|v|-22*αC(N,α)12SαN+α2(2+α)t2α*+2*α-1O(ε(N-2)α2N)|v|-22α*C(N,α)t2α*+2*α-1|β(ε)|N+α2NO(εN+αN)|v|.

For any vEλ- and t0, since aλ(v+tuε,v+tuε)aλ(tuε,tuε) and all norms on Eλ- are equivalent, from the Fubini theorem and (3.9)–(3.11), we conclude there exist C3,C4>0 such that, as ε0+,

(3.12)Iλ(v+tuε)I(tuε)+[|β(ε)|N+α2Nt22α*-1+|β(ε)|N+α2NO(εN+αN)t2α*+2*α-1+O(ε(N-2)α2N)t2α*+2*α-1+O(ε2+α2)t22*α-1]vλ+|β(ε)|N+α2Nt2α*-1vλ2*α+1-C3vλ22α*-C4vλ22*α.

Further, from (2.2), (2.3), (3.12) and the Young inequality, we obtain, for vEλ-, t>0 and ε0+,

Iλ(v+tuε)SαN+α2+αt2+min{C3,SαN+α2+α}42α*(22α*-1)(t22α*-1+t2α*+2*α-1+t22*α-1)vλ+2α*C32*α+1t2α*-1vλ2*α+1-C3vλ22α*-14(2α*)2SαN+α2+αt22α*SαN+α2+αt2+18(2α*)2SαN+α2+α(t22α*-1+t2α*+2*α-1+t22*α-1)22α*22α*-1+(22α*-2*α-1)C32(2*α+1)t22α*(2α*-1)22α*-2*α-1-14(2α*)2SαN+α2+αt22α*.

Whence, there exists some T*>0 large enough such that Iλ(v+tuε)0 for ε>0 small enough, all vEλ- and t>T*. Thereby, due to 2+α>(N-2)αN, it follows from (3.7) and (3.12) that, as ε0+,

supvEλ-,t0Iλ(v+tuε)=supvEλ-,t[0,T*]Iλ(v+tuε)supvEλ-,t[0,T*]{I(tuε)+[|β(ε)|N+α2N+|β(ε)|N+α2NO(εN+αN)+O(ε(N-2)α2N)+O(ε2+α2)]vλ+|β(ε)|N+α2Nvλ2*α+1-C3vλ22α*-C4vλ22*α}supt0I(tuε)+supvEλ-{[|β(ε)|N+α2Nvλ-C3vλ22α*]+[O(ε(N-2)α2N)vλ-C42vλ22*α]+[|β(ε)|N+α2Nvλ2*α+1-C42vλ22*α]}maxt0I(tuε)+|β(ε)|(N+α)2N(N+2α+2)+O(ε(N-2)(N+α)αN(N+2α)).

Since (N-2)(N+α)αN(N+2α)>2*α and (3.7) implies limε0+[ε2*α|β(ε)|-(N+α)2N(N+2α+2)]=+, repeating the proof of Lemma 2.2, we conclude (3.8) holds for ε>0 small enough. Thus this lemma is proved. ∎

Next, similar to Lemma 2.3, we give some properties of (Ce)c sequence for the functional Iλ.

Lemma 3.5

There exists Λ¯Λ¯1 such that, for any λΛ¯ and (Ce)c sequence {un} of the functional Iλ at level c(0,m*), up to a subsequence, unu in Eλ with 𝑢 being a nonzero critical point of Iλ.

Proof

Fix λΛ¯1 initially. As a starting point, we assert {un} is bounded in Eλ. If not, {un}Eλ\{0} and unλn up to a subsequence. Setting vn=ununλ, there is vEλ such that, up to a subsequence,

(3.13)vnvinEλandvn(x)v(x)a.e. inRNasn.

Before concluding contradictions in the cases of v=0 and v0, from the definition of {un}, we obtain

(3.14)m*+o(1)Iλ(un)-12Iλ(un),un=(N-2)(2+α)2(N+α)2RN(Iα|un|2α*)|un|2α*dx+Nα2(N+α)2RN(Iα|un|2*α)|un|2*αdx+Nα+N-α(N+α)2RN(Iα|un|2α*)|un|2*αdx.

If v=0, as a consequence of (3.13), there holds vn20 in LNN-2(RN). Then it follows from (V1) that vnn0 in L2({xRN:V(x)<0}). Thereby, from (3.14) and the fact |V-|<, we conclude that

Iλ(un)unλ2=12-λ2RNV-(x)vn2dx-12unλ2RN(Iα(|un|2α*2α*+|un|2*α2*α))(|un|2α*2α*+|un|2*α2*α)dx12

as n, which contradicts

limnIλ(un)unλ2=0.

If v0, there is some σ>0 such that the set Ω1={xRN:|v(x)|σ} satisfies |Ω1|>0. Then, dividing both sides of (3.14) by unλ2, we have

0(N-2)(2+α)Aα2(N+α)2lim infnΩ1Ω1|un(x)|2α*-1|vn(x)||un(y)|2α*-1|vn(y)||x-y|N-αdxdy=+,

which is also a contradiction. Hence, our claim is true. That is, the sequence {un} is bounded in Eλ.

Secondly, we certify that, up to a subsequence, unu in Eλ with u0 satisfying Iλ(u)=0 in Eλ* for λ>0 large enough. Since {un} is bounded in Eλ, there is uEλ such that, up to subsequences,

(3.15){unuinEλ,unuinLlocs(RN)for anys[1,2*),un(x)u(x)a.e. inRN.

It is standard to conclude from (3.15) that Iλ(u)=0 in Eλ*. We suppose inversely that u=0. Define

η~=lim supnsupzRNB1(z)un2dx.

It is clear that η~[0,+). If η~=0, the Lions lemma (see [44, Lemma 1.21]) implies that un0 in Ls(RN) for all s(2,2*). Thereby, it follows from (V1), (V2), the Hölder and H-L-S inequalities that

(3.16)limn{xRN:V(x)M}un2dx=0,
(3.17)limn{xRN:V(x)M}V(x)un2dx=0,
(3.18)limnRN(Iα|un|2α*)|un|2*αdx=0.
As a consequence, we deduce from the fact Iλ(un),un=o(1), (3.17) and (3.18) that, for any λΛ¯1,

(3.19)RN|un|2dx+λ{xRN:V(x)>M}V(x)un2dx=12α*RN(Iα|un|2α*)|un|2α*dx+12*αRN(Iα|un|2*α)|un|2*αdx+o(1).

Moreover, for any fixed λΛ¯1, it follows from the premise Iλ(un)nc and (3.17)–(3.19) that

(3.20)c+o(1)=2α*-12(2α*)2RN(Iα|un|2α*)|un|2α*dx+2*α-12(2*α)2RN(Iα|un|2*α)|un|2*αdx.

Take Λ¯2max{Λ¯1,(MS¯α)-1(NN+α)N-αN+α(2Nm*α)αN+α}, and further let λΛ¯2. With limit formula (3.19) in hand, we obtain from (3.16) as well as the definitions of S¯α,Sα given in (1.7) and (1.8) that

(3.21)12α*RN(Iα|un|2α*)|un|2α*dx+12*αRN(Iα|un|2*α)|un|2*αdx+o(1)RN|un|2dx+Λ¯2M{xRN:V(x)>M}un2dx+o(1)Sα(RN(Iα|un|2α*)|un|2α*dx)12α*+Λ¯2MS¯α(RN(Iα|un|2*α)|un|2*αdx)12*α.

Set

lim supnRN(Iα|un|2α*)|un|2α*dx=κ1andlim supnRN(Iα|un|2*α)|un|2*αdx=κ2.

Clearly, κ1,κ2[0,+). Passing to the limit as n in (3.21), we conclude the following four possibilities:

(1)κ1=κ2=0;
(2)κ1=0andκ2(Λ¯22*αMS¯α)2*α2*α-1;
(3)κ1(2α*Sα)2α*2α*-1andκ2=0;
(4)κ1(2α*Sα)2α*2α*-1orκ2(Λ¯22*αMS¯α)2*α2*α-1ifκ1κ20.
In case (1), by (3.20), we obtain c=0, which contradicts the premise c>0. Analogously, in each case of (2)–(4), from (3.20) again, we have cm*, which is impossible due to the premise c<m*.

We next turn to the case of η~>0. According to (3.15), there holds un20 in LNN-2(RN). Then it follows from (V1) and (V2) that RNV-(x)un2dx0. In view of this fact, for any λΛ¯1, we obtain

m*+o(1)Iλ(un)-122*αIλ(un),unα2(N+α)unλ2+o(1).

Consequently, there exists some C5>0, independent of 𝜆, such that

lim supnunλC5for anyλ[Λ¯1,+).

By (V1), there exists R¯>0 large enough such that |{xBR¯c:V(x)M}|<(η~S4C52)N2. Then the Hölder inequality and the definition of best Sobolev constant 𝑆 imply that, for any λΛ¯1,

(3.22)lim supn{xBR¯c:V(x)M}un2dx|{xBR¯c:V(x)M}|2NS-1lim supnunλ2η~4.

Moreover, taking Λ¯3=max{Λ¯1,4C52η~M} and letting λΛ¯3, by the fact that lim supnunλC5, we have

(3.23)lim supn{xBR¯c:V(x)>M}un2dx1λMlim supnunλ2C52λMη~4.

Then, combining (3.15), (3.22), (3.23) and the boundedness of {|un|2}, we obtain that, for any λΛ¯3,

η~=lim supnsupzRNB1(z)un2dxlim supn(BR¯+{xBR¯c:V(x)M}+{xBR¯c:V(x)>M})un2dxη~2,

a contradiction. Now, setting Λ¯=max{Λ¯2,Λ¯3}, we deduce from the above arguments that unu in Eλ up to a subsequence with u0 satisfying Iλ(u)=0 for any λΛ¯. Thus this lemma is proved. ∎

Proof of Theorem 1.2

Let λΛ¯. Based on the linking theorem [34, Theorem 5.3], we deduce from Lemmas 3.3 and 3.4 that the functional Iλ has a (Ce)cλ sequence in Eλ with cλ(0,m*). Then, due to Lemma 3.5, equation (1.5) has a nontrivial solution. Thus the proof of Theorem 1.2 is completed. ∎

Award Identifier / Grant number: 11971393

Award Identifier / Grant number: XDJK2020D032

Funding statement: Supported by National Natural Science Foundation of China (No. 11971393) and Fundamental Research Funds for the Central Universities (No. XDJK2020D032).

Acknowledgements

The authors would like to thank the anonymous referee for carefully reading this paper and making valuable comments and suggestions which greatly improve the original manuscript.

  1. Communicated by: Zhi-Qiang Wang

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Received: 2020-06-11
Revised: 2020-09-16
Accepted: 2020-10-05
Published Online: 2020-11-04
Published in Print: 2021-02-01

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