Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2018: 3.18

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …

Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth

Daniele CassaniORCID iD: https://orcid.org/0000-0002-2672-853X / Jianjun Zhang
  • College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P. R. China; and Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, via G.B. Vico 46, 21100 Varese, Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-06-07 | DOI: https://doi.org/10.1515/anona-2018-0019

Abstract

We are concerned with the existence of ground states and qualitative properties of solutions for a class of nonlocal Schrödinger equations. We consider the case in which the nonlinearity exhibits critical growth in the sense of the Hardy–Littlewood–Sobolev inequality, in the range of the so-called upper-critical exponent. Qualitative behavior and concentration phenomena of solutions are also studied. Our approach turns out to be robust, as we do not require the nonlinearity to enjoy monotonicity nor Ambrosetti–Rabinowitz-type conditions, still using variational methods.

Keywords: Ground states; semiclassical states; Choquard equation; Hardy–Littlewood–Sobolev inequality; upper-critical exponent

MSC 2010: 35B25; 35B33; 35J61

1 Introduction and main results

This paper deals with the following class of nonlinear and nonlocal Schrödinger equations:

-ε2Δv+V(x)v=ε-α(IαF(v))f(v),v>0,xN,(1.1)

where ε>0 is the dimensionalized Planck constant, N3, α(0,N), F is the primitive function of f, Iα is the Riesz potential defined for every xN{0} by

Iα(x):=Aα|x|N-α,where Aα=Γ(12(N-α))Γ(α2)πN22α,Γ is the Gamma function,

and the external potential V satisfies:

  • (V1)

    VC(N,) and infxNV(x)>0.

When ε=1, V(x)=a>0, equation (1.1) reduces to the following nonlocal elliptic equation:

-Δu+au=(IαF(u))f(u),u>0,xN,(1.2)

which is variational, in the sense that solutions of (1.2) turn out to be critical points of the energy functional

La(u)=12N|u|2+au2-(IαF(u))F(u),uH1(N).

In particular, in the relevant physical case of dimension N=3, α=2 and F(s)=s22, (1.2) turns into the so-called Choquard equation

-Δu+au=(I2u2)u,x3,(1.3)

which goes back to the seminal work of Fröhlich [24] and Pekar [50], modeling the quantum Polaron and then used by Choquard [35] to study steady states of the one component plasma approximation in the Hartree–Fock theory [38]. Equation (1.3) appears also in quantum gravity in the form of Schrödinger–Newton systems [51, 52, 53] in which a single particle is moving in its own gravitational field (self-gravitating matter), see also [30]. Lieb in [35] proved the existence and uniqueness of positive solutions to (1.3) by using rearrangements techniques. Multiplicity results for (1.3) were then obtained by Lions [39, 40] by means of a variational approach. A class of solutions which turn out to be of great interest in Physics as well as Mathematics are minimal energy solutions, which were predicted by Pekar to have a stochastic characterization in terms of Brownian motion, a conjecture proved just thirty years later by Donsker and Varadhan [21, 22]. We refer to [47] and references therein for an extensive survey on the topic.

Set ε=1, V1 and F(u)=|u|pp in (1.3):

-Δu+u=(Iα|u|p)|u|p-2u,xN.(1.4)

Formally, as α0, equation (1.4) yields

-Δu+u=|u|2p-2u,xN,(1.5)

which is a prototype in semilinear equations and in particular it is well known since the work of Gidas, Ni and Nirenberg [28] that positive solutions with finite energy are radially symmetric, unique and non-degenerate (in the sense that the kernel of the linearized operator at the solution u is generated by u), see [28, 49]. In contrast with the local problem (1.5), moving planes methods are somehow difficult to be used and is difficult to be used and the classification of positive solutions to (1.4) (even for p=2) has remained open for a long time. By using a suitable version of the moving planes method developed by Chen, Li and Ou [15], Ma and Zhao [42] gave a breakthrough to this open problem by considering equivalent Bessel–Riesz integral systems. By requiring some involved assumptions on α, p and N, they proved that positive solutions of (1.4) are, up to translations, radially symmetric and unique. In [44], Moroz and Van Schaftingen established the existence of ground state solutions to (1.4) in the optimal range

N+αN<p<N+αN-2.(1.6)

The endpoints in the above range of p are extremal values for the Hardy–Littlewood–Sobolev inequality [36] and sometimes called lower and upper H-L-S critical exponents. From the PDE point of view, a Pohozaev-type identity prevents the existence of finite energy solutions. In the upper critical case, as in the local Sobolev case, the appearance of a group invariance which yields explicit extremal functions to the H-L-S inequality is responsible for the lack of compactness. The lack of compactness can not be recovered by the presence of an external potential. In the lower critical case, equivalent variational characterizations of the ground state level still allow the H-L-S extremal functions to preventing compactness: this casts the problem within the class of Brezis–Nirenberg-type problems [8].

Recently, in [45] the more general Choquard equation (1.2) has been studied by requiring Berestycki–Lions-type conditions, and establishing the existence of ground state solutions in the subcritical case (1.6).

The first purpose of the present work is to investigate the existence of ground state solutions to (1.2) involving the upper H-L-S critical exponent. In presence of lower H-L-S critical exponent, a suitable external potential may lower down the groundstate to the compact region. This turns out to be a Lions-type problem and it is considered in a companion paper [26] as it involves quite different techniques.

Definition 1.1.

A function u is said to be a ground state solution of (1.2) if u is a solution of (1.2) with the least action energy among all nontrivial solutions of (1.2). Namely,

La(u)=inf{La(v):vH1(N) is a solution to (1.2)}.

Throughout this paper we assume fC(+,) which satisfies:

  • (F1)

    limt0+f(t)t=0,

  • (F2)

    limt+f(t)t-α+2N-2=1,

  • (F3)

    there exist μ>0 and q(2,N+αN-2) such that

    f(t)t2+αN-2+μtq-1,t>0.

Our first main result in this paper is the following:

Theorem 1.1.

Assume α((N-4)+,N), q>max{1+αN-2,N+α2(N-2)} and (F1)(F3). Then, for any a>0, (1.2) admits a ground state solution.

Let us point out that assumption (F3) plays a crucial role. Indeed, under the lonely assumptions (F1) and (F2), equation (1.2) has no solutions for any nontrivial external potential V by means of a Pohozaev-type identity (Lemma 3.2, Section 2). This fact rules out any perturbative argument and casts the problem into a Brezis–Nirenberg-type.

The second purpose of this paper is to investigate the profile of positive solutions to (1.1) as ε0. Indeed, in quantum physics one expects that as the Planck constant ε0, the dynamic is governed by the external potential V and an interesting class of solutions show up which develop a spike shape around critical points of V. From the physical point of view, these solutions are known as semiclassical states, as they describe the transition from quantum mechanics to classical mechanics. For the detailed physical background, we refer to [49] and references therein. By a Lyapunov–Schmidt reduction approach, based on the non-degeneracy condition, in [23, 49] the authors obtained the existence of solutions to the semilinear singularly perturbed Schrödinger equation

-ε2Δu+V(x)u=f(u),(1.7)

which exhibit a single peak or multi peaks concentrating, as ε0, around any given non-degenerate critical points of V. However, so far, the non-degeneracy condition holds for only a very restricted class of f. In the last decade, a lot of efforts have been devoted to relax or remove the non-degeneracy condition in this family of singularly perturbed problems. By using a variational approach, Rabinowitz [54] obtained the existence of positive solutions to (1.7) for small ε>0 with the following global potential well condition:

lim inf|x|V(x)>infNV(x).

Subsequently, by a penalization approach, del Pino and Felmer [18] weakened the above global potential well condition to the local condition

  • (V2)*

    there exists a bounded domain ON such that

    0<minfxOV(x)<minxOV(x)

and proved the existence of a single-peak solution to (1.7). In [54, 18], the non-degeneracy condition is not required. Some related results can be found in [59, 19, 20, 17, 3] and the references therein. In [10] Byeon and Jeanjean introduced a new penalization approach and constructed a spike layered solution of (1.7) under (V2)* and the almost optimal Berestycki-Lions conditions [6], see also [11, 12, 9] and [65, 68].

The second main result of this paper is the following:

Theorem 1.2.

Assume (V1)(V2)* in addition to the assumptions of Theorem 1.1 and let M{xO:V(x)=m}. Then, for small ε>0, (1.1) admits a positive solution vε, which satisfies:

  • (i)

    There exists a local maximum point xε of vε such that

    limε0dist(xε,)=0,

    and wε(x)vε(εx+xε) converges (up to a subsequence) uniformly to a ground state solution of the limit equation

    -Δu+mu=(IαF(u))f(u),u>0,uH1(N).

  • (ii)

    vε(x)Cexp(-cε|x-xε|) for some c,C>0.

We mention that related results under stronger assumptions have been recently obtained in [5]. For the convenience of the reader let us better contextualize our result within the existing literature on the singularly perturbed problem (1.1).

In [60], Wei and Winter considered the nonlocal equation, equivalent to the Schrödinger–Newton system,

-ε2Δv+V(x)v=ε-2(I2v2)v,x3,(1.8)

and by using a Lyapunov–Schmidt reduction method under assumption (V1), proved the existence of multi-bump solutions concentrating around local minima, local maxima or non-degenerate critical points of V. When the potential is allowed to vanish somewhere, thus avoiding (V1), the problem becomes much more difficult. In [56], Secchi considered (1.8) with a positive decaying potential and by means of a perturbative approach, proved the existence and concentration of bound states near local minima (or maxima) points of V as ε0. Recently, by a nonlocal penalization technique, Moroz and Van Schaftingen [46] obtained a family of single spike solutions for the Choquard equation

-ε2Δv+V(x)v=ε-α(Iα|v|p)|v|p-2v,xN,

around the local minimum of V as ε0. In [46] the assumption on the decay of V and the range for p2 are optimal. More recently, using the penalization argument introduced in [10], Yang, Zhang and Zhang [64] investigated the existence and concentration of solutions to (1.1) under the local potential well condition (V2)* and mild assumptions on f. In particular, the Ambrosetti–Rabinowitz condition and the monotonicity of f(t)t are not required. For related results see [4, 58, 16, 43, 48, 56, 63, 7]. All the previous results are subcritical in the sense of the Hardy–Littlewood–Sobolev inequality. In [2], the authors considered the ground state solutions of the Choquard equation (1.1) in 2. By variational methods, the authors proved the existence and concentration of ground states to (1.1) involving critical exponential growth in the sense of the Pohozaev–Trudinger–Moser inequality. A natural open problem which has not been settled before is to establish concentration phenomena for (1.1) in the critical growth regime. Here we give a positive answer to this open problem in Theorem 1.2.

Overview.

We conclude this section by giving the outline of the paper and pointing out major difficulties. In Section 2 we prove some preliminary results which require some efforts to extend a few well-known results in the local setting, to the nonlocal framework. Section 3 is devoted to proving Theorem 1.1. Here, without the Ambrosetti–Rabinowitz condition, to obtain the boundedness of the Palais–Smale sequence becomes a delicate issue. To overcome this difficulty, a possible strategy is to look for a constraint minimization problem. This goes back to Berestycki–Lions [6], in which the authors established the existence of ground state solutions to the scalar mean field equation -Δu=g(u),uH1(N). By using a similar strategy, Zhang and Zou [67] extended the result in [6] to the critical case. Precisely, in [6, 67], the existence of ground state solutions is reduced to looking at the constraint minimization problem

inf{12N|u|2:NG(u)=1,uH1(N)}

and eventually to get rid of the Lagrange multiplier thanks to some appropriate scaling. However, this approach fails for the nonlocal problem (1.2), since N|u|2, N|u|2 and N(IαF(u))F(u) scale differently in space and hence one has no hope to remove the Lagrange multiplier. The existence of ground state solutions to the nonlocal problem (1.2), in the subcritical case, has been done by Moroz and Van Schaftingen in [45], where they constructed a bounded Palais–Smale sequence satisfying asymptotically the Pohozaev identity and obtained a ground state solution by virtue of a concentration-compactness-type argument and a scaling technique introduced by Jeanjean [31]. Here, to avoid a Ambrosetti–Rabinowitz-type condition, we use the Struwe monotonicity trick, in the abstract form due to [32], to get a bounded Palais–Smale sequence. Clearly, due to the presence of a critical H-L-S term, the Palais–Smale condition fails. By a decomposition technique, we recover compactness and obtain the existence of ground state solutions to (1.2). In Section 4, we first prove some qualitative properties of the set of ground states such as compactness, regularity, symmetry and positivity. Then we use a truncation argument as key ingredient to prove Theorem 1.2. In [64], the authors considered problem (1.1) in the subcritical case and established concentration phenomena. Here, the presence of critical growth prevents to use directly the argument in [64]. We overcome this difficulty by penalizing the problem which is relaxed to a subcritical case. The penalized problem admits a family of spike shaped solutions which develop a concentrating behavior around the local minima of V. Finally, the analysis carried out in Section 3 enables us to prove the convergence of the penalized solution to a solution of the original problem which preserves the same qualitative properties of the penalized problem.

2 Preliminaries

In this section, we are concerned with the existence of ground state solutions to (1.2). Let a>0 and denote the least energy of (1.2) by

Ea=inf{La(u):La(u)=0 in H-1(N),uH1(N){0}}.

In what follows, let H1(N) be endowed with the norm

u=(N|u|2+a|u|2)12,uH1(N).

Before proving Theorem 1.1, we prove first some preliminary results. First of all, let us recall the following Hardy–Littlewood–Sobolev inequality which will be frequently used throughout the paper.

Lemma 2.1 ([37, Theorem 4.3]).

Let s,r>1 and 0<α<N with 1s+1r=1+αN, fLs(RN) and gLr(RN). Then there exists a positive constant C(s,N,α) (independent of f,g) such that

|NNf(x)|x-y|α-Ng(y)dxdy|C(s,N,α)fsgr.

In particular, if s=r=2NN+α, the best possible constant is given by

𝒞α:=πN-α2Γ(α2)Γ(N+α2)[Γ(N2)Γ(N)]-αN.

Remark 2.1.

As a consequence of the Hardy–Littlewood–Sobolev inequality, for any vLs(N), s(1,Nα), IαvLNs/(N-αs)(N). Moreover, Iα(Ls(N),LNs/(N-αs)(N)) and

IαvNsN-αsC(s,N,α)vs.

2.1 Brezis–Lieb lemma

In this subsection, we prove a nonlocal version of the Brezis–Lieb lemma.

Lemma 2.2 (Brezis–Lieb Lemma).

Assume α(0,N) and there exists a constant C>0 such that

|f(t)|C(|t|αN+|t|α+2N-2),s.

Let {un}H1(RN) be such that unu weakly in H1(RN) and a.e. in RN as n. Then

N(IαF(un))F(un)=N(IαF(un-u))F(un-u)+N(IαF(u))F(u)+on(1),

where on(1)0 as n.

In order to prove Lemma 2.2, we recall the following lemma, which states that pointwise convergence of a bounded sequence implies weak convergence.

Lemma 2.3 ([62, Theorem 4.2.7]).

Let ΩRN be a domain and let {un} be bounded in Lq(Ω) for some q>1. If unu a.e. in Ω as n, then unu weakly in Lq(Ω) as n

Proof of Lemma 2.2.

Observe that

N(IαF(un))F(un)-(IαF(un-u))F(un-u)-(IαF(u))F(u)=N(Iα[F(un)+F(un-u)])[F(un)-F(un-u)]-(IαF(u))F(u)

and there exists C>0 such that

|F(s)|C(|s|N+αN+|s|N+αN-2)for all s,

which implies F(u)L2N/(N+α)(N). For any δ>0 sufficiently small, by the Hardy–Littlewood–Sobolev inequality there exists K1>0 such that

|Ω1(IαF(u))F(u)|δ6,Ω1:={xN:|u(x)|K1}.

Again by the Hardy–Littlewood–Sobolev inequality we have

|Ω1(Iα[F(un)+F(un-u)])[F(un)-F(un-u)]|C(N|F(un)+F(un-u)|2NN+α)N+α2N(Ω1|F(un)-F(un-u)|2NN+α)N+α2NC(N,α)(Ω1|F(un)-F(un-u)|2NN+α)N+α2N,

where we have used the fact that {un} is bounded in H1(N). It is easy to see there exists c>0 such that

|F(un)-F(un-u)|2NN+αc(|un|2αN+α|u|2NN+α+|un|2+αN-22NN+α|u|2NN+α+|u|2+|u|2NN-2).

Then, by Hölder’s inequality,

Ω1|un|2αN+α|u|2NN+α(Ω1|un|2)αN+α(Ω1|u|2)NN+α

and

Ω1|un|2+αN-22NN+α|u|2NN+α(Ω1|un|2NN-2)2+αN+α(Ω1|u|2NN-2)N-2N+α.

So for δ given above and K1 fixed but large enough, we get for any n,

|Ω1(Iα[F(un)+F(un-u)])[F(un)-F(un-u)]|δ6.

Similarly, let Ω2:={xN:|x|R}Ω1 with R>0 large enough, we have

|Ω2(IαF(u))F(u)|δ6

and for any n,

|Ω2(Iα[F(un)+F(un-u)])[F(un)-F(un-u)]|δ6.

For K2>K1, let Ω3(n):={xN:|un(x)|K2}(Ω1Ω2). If Ω3(n), then we know that |u(x)|<K1 and |x|<R for any xΩ3(n). By noting that unu a.e. in Ω as n, it follows from the Severini-Egoroff theorem that un converges to u in measure in BR(0), which implies that |Ω3(n)|0 as n. Similarly we have, for n large enough,

|Ω3(n)(IαF(u))F(u)|δ6

and

|Ω3(n)(Iα[F(un)+F(un-u)])[F(un)-F(un-u)]|δ6.

Finally, let us estimate

Ω4(n)(Iα[F(un)+F(un-u)])[F(un)-F(un-u)]-(IαF(u))F(u),

where Ω4(n)=N(Ω1Ω2Ω3(n)). Obviously, Ω4(n)BR(0). By Lebesgue’s dominated convergence theorem we have

limnΩ4(n)|F(un-u)|2NN+α=0, and limnΩ4(n)|F(un)-F(u)|2NN+α=0,

which implies by the Hardy–Littlewood–Sobolev inequality

|Ω4(n)(Iα[F(un)+F(un-u)])F(un-u)|C(N,α)(Ω4(n)|F(un-u)|2NN+α)N+α2N0

as n, and

|Ω4(n)(Iα[F(un)+F(un-u)])[F(un)-F(u)]|C(N,α)(Ω4(n)|F(un)-F(u)|2NN+α)N+α2N0

as n. Now let Hn=F(un)+F(un-u)-F(u); we have

limnΩ4(n)(Iα[F(un)+F(un-u)])[F(un)-F(un-u)]-(IαF(u))F(u)=limnΩ4(n)(IαHn)F(u).

Noting that Hn is bounded in L2N/(N+α)(N) and Hn0 a. e. in N as n, by Lemma 2.3, Hn0 weakly in L2N/(N+α)(N) as n. By Remark 2.1, IαHn0 weakly in L2N/(N-α)(N) as n, which yields

limnΩ4(n)(IαHn)F(u)=0.

Thus,

lim supn|N(IαF(un))F(un)-(IαF(un-u))F(un-u)-(IαF(u))F(u)|δ

and the arbitrary choice of δ concludes the proof. ∎

2.2 Splitting lemma

Next we prove a splitting property for the nonlocal energy.

Lemma 2.4 (Splitting Lemma).

Assume α((N-4)+,N), (F1)(F2) and let {un}H1(RN) be such that unu weakly in H1(RN) and a.e. in RN as n. Then, up to a subsequence if necessary,

N([IαF(un)]f(un)-[IαF(un-u)]f(un-u)-[IαF(u)]f(u))ϕ=on(1)ϕ,

where on(1)0 uniformly as n for any ϕC0(RN).

In order to prove Lemma 2.4, we need first to prove Lemma 2.5 and Lemma 2.6 below.

Lemma 2.5.

Let ΩRN be a domain and let {un}H1(Ω) be such that unu weakly in H1(Ω) and a.e. in Ω as n. Then the following hold:

  • (i)

    For any 1<qr2NN-2 and r>2,

    limnΩ||un|q-1un-|un-u|q-1(un-u)-|u|q-1u|rq=0.

  • (ii)

    Assume hC(,) and h(t)=o(t) as t0, |h(t)|c(1+|t|q) for any t , where q(1,N+2N-2] . The following hold:

    • (ii)(2)

      For any r[q+1,2NN-2],

      limnΩ|H(un)-H(un-u)-H(u)|rq+1=0.

      where H(t)=0th(s)ds,

    • (ii)(2)

      If we further assume that Ω=N, α((N-4)+,N) and lim|t|h(t)|t|-α+2N-2=0 , then

      N|h(un)-h(un-u)-h(u)|2NN+α|ϕ|2NN+α=on(1)ϕ2NN+α,

      where on(1)0 uniformly for any ϕC0(N) as n.

Proof.

The proofs of (i) and (1) are similar to [66, Lemma 2.5]. We only give the proof of (2) which is inspired by [1] and [68, Lemma 4.7].

In the following, let C denote a positive constant (independent of ε,k) which may change from line to line. For any fixed ε(0,1), there exists s0=s0(ε)(0,1) such that |h(t)|ε|t| for |t|2s0. Choose s1=s1(ε)>2 such that

|h(t)|ε|t|2+αN-2

for |t|s1-1. From the continuity of h, there exists δ=δ(ε)(0,s0) such that |h(t1)-h(t2)|s0ε for |t1-t2|δ,|t1|,|t2|s1+1. Moreover, there exists c(ε)>0 such that

|h(t)|c(ε)|t|+ε|t|2+αN-2

for t. Noting that α((N-4)+,N), we have 2<4NN+α<2NN-2. Then there exists R=R(ε)>0 such that

NB(0,R)|h(u)ϕ|2NN+αCNB(0,R)(|u|2NN+α+ε|u|2+αN-22NN+α)|ϕ|2NN+αC(NB(0,R)|u|4NN+α)12(N|ϕ|4NN+α)12+Cε(NB(0,R)|u|2NN-2)2+αN+α(N|ϕ|2NN-2)N-2N+αCεϕ2NN+α.(2.1)

Setting An:={xNB(0,R):|un(x)|s0}; then

An{|u|δ}|h(un)-h(un-u)|2NN+α|ϕ|2NN+αCεN(|un|2NN+α+|un-u|2NN+α)|ϕ|2NN+αCεϕ2NN+α.

Let Bn:={xNB(0,R):|un(x)|s1}. Then

Bn{|u|δ}|h(un)-h(un-u)|2NN+α|ϕ|2NN+αCεN(|un|2+αN-22NN+α+|un-u|2+αN-22NN+α)|ϕ|2NN+αCεϕ2NN+α.

Setting Cn:={xNB(0,R):s0|un(x)|s1}; then |Cn|< and

Cn{|u|δ}|h(un)-h(un-u)|2NN+α|ϕ|2NN+α(s0ε)2NN+αCn{|u|δ}|ϕ|2NN+α(s0ε)2NN+α|Cn|12(N|ϕ|4NN+α)12ε2NN+α(Cn|un|4NN+α)12(N|ϕ|4NN+α)12Cεϕ2NN+α.

Thus, (NB(0,R)){|u|δ}=AnBnCn and

(NB(0,R)){|u|δ}|h(un)-h(un-u)|2NN+α|ϕ|2NN+αCεϕ2NN+αfor all n.

Clearly, for ε given above, there exists c(ε)>0 such that

|h(un)-h(un-u)|2NN+αε(|un|2+αN-22NN+α+|un-u|2+αN-22NN+α)+c(ε)(|un|2NN+α+|un-u|2NN+α)

and

(NB(0,R)){|u|δ}|h(un)-h(un-u)|2NN+α|ϕ|2NN+α(NB(0,R)){|u|δ}ε(|un|2+αN-22NN+α+|un-u|2+αN-22NN+α)|ϕ|2NN+α+c(ε)(|un|2NN+α+|un-u|2NN+α)|ϕ|2NN+αCεϕ2NN+α+c(ε)(NB(0,R)){|u|δ}(|un|2NN+α+|un-u|2NN+α)|ϕ|2NN+α.

Noting that 0<α+4-N<N+α and |(NB(0,R)){|u|δ}|0 as R, there exists R=R(ε) large enough, such that

(NB(0,R)){|u|δ}c(ε)(|un|2NN+α+|un-u|2NN+α)|ϕ|2NN+αc(ε)[(N|un|2NN-2)N-2N+α+(N|un-u|2NN-2)N-2N+α](N|ϕ|2NN-2)N-2N+α|(NB(0,R)){|u|δ}|α+4-NN+αεϕ2NN+α.

Then, for any n,

(NB(0,R)){|u|δ}|h(un)-h(un-u)|2NN+α|ϕ|2NN+αCεϕ2NN+α.

Thus, by (2.1), for any n,

NB(0,R)|h(un)-h(u)-h(un-u)|2NN+α|ϕ|2NN+αCεϕ2NN+α.(2.2)

Finally, for ε>0 given above, there exists C(ε)>0 such that

|h(t)|2NN+αC(ε)|t|2NN+α+ε|t|2NN+α2+αN-2,t.(2.3)

Recalling that unu weakly in H1(N), up to a subsequence, unu strongly in L4N/(N+α)(B(0,R)) and there exists ωL4N/(N+α)(B(0,R)) such that |un(x)|,|u(x)||ω(x)| a.e. xB(0,R). Then we easily get for n large enough,

B(0,R)|h(un-u)|2NN+α|ϕ|2NN+αB(0,R)(C(ε)|un-u|2NN+α+ε|un-u|2NN+α2+αN-2)|ϕ|2NN+αCεϕ2NN+α.(2.4)

Moreover, let Dn:={xB(0,R):|un(x)-u(x)|1}, then by (2.3),

Dn|h(un)-h(u)|2NN+α|ϕ|2NN+αDn[C(ε)(|u|2NN+α+|un|2NN+α)+ε(|un|2NN+α2+αN-2+|u|2NN+α2+αN-2)]|ϕ|2NN+αCεϕ2NN+α+2C(ε)Dn|ω|2NN+α|ϕ|2NN+αCεϕ2NN+α+2C(ε)(Dn|ω|4NN+α)12(N|ϕ|4NN+α)12.

By unu a.e. xB(0,R), we get |Dn|0 as n. Hence,

Dn|h(un)-h(u)|2NN+α|ϕ|2NN+αCεϕ2NN+αfor n large.(2.5)

On the other hand, for ε given above, there exists c(ε)>0 such that

|h(un)-h(u)|2NN+αε(|un|2+αN-22NN+α+|un|2+αN-22NN+α)+c(ε)(|un|2NN+α+|un|2NN+α).

Noting that |{|u|L}|0 as L, similarly as above, there exists L=L(ε)>0 such that for all n,

(B(0,R)Dn){|u|L}|h(un)-h(u)|2NN+α|ϕ|2NN+αCεϕ2NN+α.

By the Lebesgue dominated convergence theorem,

(B(0,R)Dn){|u|L}|h(un)-h(u)|2NN+α|ϕ|2NN+α=on(1)ϕ2NN+α,

where on(1)0 as n uniformly in ϕ. Then by (2.5),

B(0,R)|h(un)-h(u)|2NN+α|ϕ|2NN+αCεϕ2NN+αfor n large.

Then, by (2.4) and for n large,

B(0,R)|h(un)-h(u)-h(un-u)|2NN+α|ϕ|2NN+αCεϕ2NN+αfor n sufficiently large.

Finally, combining the previous estimate with (2.2), we conclude the proof. ∎

Lemma 2.6.

Let α(0,N), s(1,Nα) and let {gn}L1(RN)Ls(RN) be bounded and such that, up to a subsequence, for any bounded domain ΩRN, gn0 strongly in Ls(Ω) as n. Then, up to a subsequence if necessary, (Iαgn)(x)0 a.e. in RN as n.

Proof.

Let us prove that for any fixed positive k, passing to a subsequence if necessary, (Iαgn)(x)0 a.e. in Bk(0) as n. Let k+ be fixed and for any δ>0, there exists K=K(δ)>k such that

AαNBK(x)|gn(y)||x-y|N-αdyδfor any xN,n+.

Obviously, BK(x)B2K(0) for any xBK(0). Noting that gnχB2K(0)Ls(N), by Remark 2.1,

Iα(|gn|χB2K(0))LNsN-αs(N)CgnLs(B2K(0)),

where the constant C depends only on N,α. It follows that, up to a subsequence, Iα(|gn|χB2K(0))0 strongly in LNs/(N-αs)(N) and a.e. in Bk(0) as n. Then, for almost every xBk(0), one has

lim supn|(Iαgn)(x)|Aαlim supn(BK(x)|gn(y)||x-y|N-αdy+NBK(x)|gn(y)||x-y|N-αdy)δ+Aαlim supnBK(x)|gn(y)||x-y|N-αdyδ+Aαlim supnB2K(0)|gn(y)||x-y|N-αdy=δ+lim supn[Iα(|gn|χB2K(0))](x)=δ.

Since δ is arbitrary, the proof is completed. ∎

Now we are set to prove Lemma 2.4.

Proof of Lemma 2.4.

Set

f1(t)=f(t)-|t|4+α-NN-2tandF1(t)=0tf1(s)ds,t.

Observe that for any ϕC0(N),

N[IαF(un)]f(un)ϕ=N[IαF(un)]f1(un)ϕ+N[IαF(un)]|un|4+α-NN-2unϕ.

Step 1. We claim

N[IαF(un)]|un|4+α-NN-2unϕ=N[IαF(un-u)]|un-u|4+α-NN-2(un-u)ϕ+N[IαF(u)]|u|4+α-NN-2uϕ+on(1)ϕ,

where on(1)0 uniformly for any ϕC0(N) as n. Noting that α>N-4, by Lemma 2.5 (ii) (1) with h(t)=f(t), q=2+αN-2,r=2NN-2,

limnN|F(un)-F(un-u)-F(u)|2NN+α=0.(2.6)

Then for vn=|un|4+α-NN-2un, as well as vn=|un-u|4+α-NN-2(un-u) and also vn=|u|4+α-NN-2u, there exists C>0 such that

N|vnϕ|2NN+α(N|vn|2N2+α)2+αN+α(N|ϕ|2NN-2)N-2N+αC(N|ϕ|2NN-2)N-2N+α,

from which it follows

|N[Iα(F(un)-F(un-u)-F(u))]vnϕ|C(N|F(un)-F(un-u)-F(u))|2NN+α)N+α2N(N|vnϕ|2NN+α)N+α2N=on(1)(N|vnϕ|2NN+α)N+α2N=on(1)ϕ,

where on(1)0 uniformly for any ϕC0(N) as n.

On the other hand, by virtue of (i) of Lemma 2.5 with q=2+αN-2 and r=2NN-2,

limnN||un|4+α-NN-2un-|un-u|4+α-NN-2(un-u)-|u|4+α-NN-2u|2N2+α=0.

For wn=F(un), as well as wn=F(un-u) and also wn=F(u), one easily gets {wn} bounded in L2N/(N+α)(N). By the Hardy–Littlewood–Sobolev inequality and Hölder’s inequality, there exists C>0 such that

|N[Iαwn][|un|4+α-NN-2un-|un-u|4+α-NN-2(un-u)-|u|4+α-NN-2u]ϕ|C(N||un|4+α-NN-2un-|un-u|4+α-NN-2(un-u)-|u|4+α-NN-2u|2NN+α|ϕ|2NN+α)N+α2NC(N||un|4+α-NN-2un-|un-u|4+α-NN-2(un-u)-|u|4+α-NN-2u|2N2+α)2+α2N(N|ϕ|2NN-2)N-22N=on(1)ϕ,

where on(1)0 uniformly for any ϕC0(N) as n. Then we get

N[IαF(un)]|un|4+α-NN-2unϕ=N[IαF(un-u)]|un-u|4+α-NN-2(un-u)ϕ+N[IαF(u)]|u|4+α-NN-2uϕ+N[IαF(un-u)]|u|4+α-NN-2uϕ+N[IαF(u)]|un-u|4+α-NN-2(un-u)ϕ+on(1)ϕ,

where on(1)0 uniformly for any ϕC0(N) as n. Noting that F(u)L2N/(N+α)(N), by Remark 2.1, |IαF(u)|2NN+2L(N+2)/(N-α)(N). By virtue of Lemma 2.3, |un-u|2N(2+α)(N-2)(N+2)0 weakly in L(N+2)/(2+α)(N) as n0. This yields

limnN|IαF(u)|2NN+2|un-u|2N(2+α)(N-2)(N+2)=0,

which implies that

|N[IαF(u)]|un-u|4+α-NN-2(un-u)ϕ|(N|IαF(u)|2NN+2|un-u|2N(2+α)(N-2)(N+2))N+22N(N|ϕ|2NN-2)N-22N=on(1)ϕ,

where on(1)0 uniformly for any ϕC0(N) as n.

At the same time, since α((N-4)+,N), for s(1,2NN+α)(1,Nα), by Rellich’s theorem, up to a subsequence, for any bounded domain ΩN, F(un-u)0 strongly in Ls(Ω) as n. By Lemma 2.6, up to a subsequence, IαF(un-u)0 a.e. in N as n0. By Remark 2.1 we have

supn|IαF(un-u)|2NN+2L(N+2)/(N-α)(N)CsupnF(un-u)L2N/(N+α)(N)<,

which yields, by Lemma 2.3, |IαF(un-u)|2NN+20 weakly in L(N+2)/(N-α)(N) as n. Noting that |u|2+αN-22NN+2L(N+2)/(2+α)(N),

limnN|IαF(un-u)|2NN+2|u|2+αN-22NN+2=0(2.7)

and by Hölder’s inequality,

|N[IαF(un-u)]|u|4+α-NN-2uϕ|(N|IαF(un-u)|2NN+2|u|2+αN-22NN+2)N+22N(N|ϕ|2NN-2)N-22N=on(1)ϕ,

where on(1)0 uniformly for any ϕC0(N) as n. The claim is thus proved.

Step 2. We claim

N[IαF(un)]f1(un)ϕ=N[IαF(un-u)]f1(un-u)ϕ+N[IαF(u)]f1(u)ϕ+on(1)ϕ,(2.8)

where on(1)0 uniformly for any ϕC0(N) as n. The following hold:

{N[Iα(F(un)-F(un-u)-F(u))]f1(un)ϕ=on(1)ϕ,N[Iα(F(un)-F(un-u)-F(u))]f1(un-u)ϕ=on(1)ϕ,N[Iα(F(un)-F(un-u)-F(u))]f1(u)ϕ=on(1)ϕ,(2.9)

where on(1)0 uniformly for any ϕC0(N) as n. Let us only prove the first identity in (2.9), the remaining ones being similar. Observe that there exists δ(0,1) and C(δ)>0 such that |f1(t)||t| for |t|δ and |f1(t)|C(δ)|t|2+αN-2 for |t|δ. Noting that α((N-4)+,N), we have 2<4NN+α<2NN-2. Then, for any ϕC0(N), there exists C>0 (independent of ϕ,n) such that

N|f1(un)ϕ|2NN+α={xN:|un(x)|δ}|f1(un)ϕ|2NN+α+{xN:|un(x)|δ}|f1(un)ϕ|2NN+α{xN:|un(x)|δ}|unϕ|2NN+α+[C(δ)]2NN+α{xN:|un(x)|δ}|un|2N(2+α)(N-2)(N+α)|ϕ|2NN+α(N|un|4NN+α)12(N|ϕ|4NN+α)12+[C(δ)]2NN+α(N|un|2NN-2)2+αN+α(N|ϕ|2NN-2)N-2N+αCϕ2NN+αfor all n1.

Thus

(N|f1(un)ϕ|2NN+α)N+α2NCϕuniformly for all ϕC0(N),n=1,2,.

Then by the Hardy–Littlewood–Sobolev inequality and (2.6),

|N[Iα(F(un)-F(un-u)-F(u))]f1(un)ϕ|C(N|F(un)-F(un-u)-F(u)|2NN+α)N+α2N(N|f1(un)ϕ|2NN+α)N+α2N=on(1)ϕ,

where on(1)0 uniformly for any ϕC0(N) as n. So (2.9) holds.

Similarly we prove

{N(IαF(un))[f1(un)-f1(un-u)-f1(u)]ϕ=on(1)ϕ,N(IαF(un-u))[f1(un)-f1(un-u)-f1(u)]ϕ=on(1)ϕ,N(IαF(u))[f1(un)-f1(un-u)-f1(u)]ϕ=on(1)ϕ,(2.10)

where on(1)0 uniformly for any ϕC0(N) as n. By the Hardy–Littlewood–Sobolev inequality and (ii)2 of Lemma 2.5, there exists C>0 such that

|N(IαF(un))[f1(un)-f1(un-u)-f1(u)]ϕ|C(N|f1(un)-f1(un-u)-f1(u)|2NN+α|ϕ|2NN+α)N+α2N=on(1)ϕ,

where on(1)0 uniformly for any ϕC0(N) as n. So the first identity of (2.10) holds and the remaining can be proved in a similar fashion.

Combine (2.9) and (2.10) to have

N[IαF(un)]f1(un)ϕ=N[IαF(un-u)]f1(un-u)ϕ+N[IαF(u)]f1(u)ϕ+N[IαF(un-u)]f1(u)ϕ+N[IαF(u)]f1(un-u)ϕ+on(1)ϕ,

where on(1)0 uniformly for any ϕC0(N) as n. To conclude the proof of (2.8), it remains to prove

N[IαF(un-u)]f1(u)ϕ=on(1)ϕ

and

N[IαF(u)]f1(un-u)ϕ=on(1)ϕ,(2.11)

where on(1)0 uniformly for any ϕC0(N) as n. Notice that for any ε(0,1), there exist δ(0,1) and Cε>0 such that |f1(t)|ε|t| for |t|δ and |f1(t)|Cε|t|2+αN-2 for |t|δ. Then, for any ϕC0(N), by the Hardy–Littlewood–Sobolev inequality and Hölder’s inequality,

|N[IαF(un-u)]f1(u)ϕ|ε{xN:|u(x)|δ}|IαF(un-u)||uϕ|+Cε{xN:|u(x)|δ}|IαF(un-u)||u|2+αN-2|ϕ|εF(un-u)L2N/(N+α)(N)({xN:|u(x)|δ}|uϕ|2NN+α)N+α2N+Cε(N|IαF(un-u)|2NN+2|u|2+αN-22NN+2)N+22N(N|ϕ|2NN-2)N-22N.

There exists c>0 (independent of ϕ,δ,ε) such that

{xN:|u(x)|δ}|uϕ|2NN+αcϕ2NN+α.

Then by (2.7), there exists C~>0 (independent of ϕ,ε) such that

lim supn|N[IαF(un-u)]f1(u)ϕ|C~εϕ.

It follows that

N[IαF(un-u)]f1(u)ϕ=on(1)ϕ,

where on(1)0 uniformly for any ϕC0(N) as n. Similarly, (2.11) can be proved and the proof of Lemma 2.4 is complete. ∎

3 Ground state solutions: Proof of Theorem 1.1

Since we are looking for positive ground state solutions to (1.2), we may assume that f is odd in N. In this section, a key tool is a monotonicity trick, originally due to Struwe [57] and which here we borrow in the abstract form due to Jeanjean and Toland [34, 32].

For λ[12,1], we consider the following family of functionals:

Iλ(u)=12N|u|2+au2-λ2N(IαF(u))F(u),uH1(N).

Obviously, if f satisfies the assumptions of Theorem 1.1, for λ[12,1], IλC1(H1(N),) and every critical point of Iλ is a weak solution of

-Δu+au=λ(IαF(u))f(u),uH1(N).(3.1)

The existence of critical points to Iλ is a consequence of the following abstract result

Theorem A (see [32]).

Let X be a Banach space equipped with a norm X, let JR+ be an interval and let a family of C1-functionals {Iλ}λJ be given on X of the form

Iλ(u)=A(u)-λB(u),uX.

Assume that B(u)0 for any uX, at least one between A and B is coercive on X and there exist two points v1,v2X such that for any λJ,

cλ:=infγΓmaxt[0,1]Iλ(γ(t))>max{Iλ(v1),Iλ(v2)},

where Γ:={γC([0,1],X):γ(0)=v1,γ(1)=v2}. Then, for almost every λJ, the C1-functional Iλ admits a bounded Palais–Smale sequence at level cλ. Moreover, cλ is left-continuous with respect to λ[12,1].

In the following, set X=H1(N) and

A(u)=12N|u|2+au2,B(u)=12N(IαF(u))F(u).

Obviously, A(u)+ as u. Thanks to (F3), B(u)0 for any uH1(N). Moreover, by (F1)(F2), for any ε>0, there exists Cε>0 such that F(t)ε|t|(N+α)/N+Cε|t|(N+α)/(N-2) for any t. Then, as in [45], there exists δ>0 such that

N(IαF(u))F(u)12u2if uδ,

and therefore for any uH1(N) and λJ,

Iλ(u)14N|u|2+au2>0if 0<uδ.(3.2)

On the other hand, for fixed 0u0H1(N) and for any λJ,t>0, by (F3),

Iλ(λu0)t22N|u0|2+a|u0|2-t2N+2αN-24(N-2N+α)2N(Iα|u0|N+αN-2)|u0|N+αN-2

and Iλ(tu0)- as t. Then there exists t0>0 (independent of λ) such that Iλ(t0u0)<0, λJ and t0u0>δ. Let

cλ:=infγΓmaxt[0,1]Iλ(γ(t)),

where Γ:={γC([0,1],X):γ(0)=0,γ(1)=t0u0}.

Remark 3.1.

Here we remark that cλ is independent of u0. In fact, let

dλ:=infγΓ1maxt[0,1]Iλ(γ(t)),

where Γ1:={γC([0,1],X):γ(0)=0,Iλ(γ(1))<0}. Clearly, dλcλ. On the other hand, for any γΓ1, it follows from (3.2) that γ(1)>δ. Due to the path connectedness of H1(N), there exists γ~C([0,1],H1(N)) such that γ~(t)=γ(2t) if t[0,12], γ~(t)>δ if t[12,1] and γ~(1)=t0u0. Then γ~Γ and

maxt[0,1]Iλ(γ~(t))=maxt[0,1]Iλ(γ(t)),

which implies that cλdλ and so dλ=cλ for any λJ.

By (3.2), cλ>δ24 for any λJ. Then, as a consequence of Theorem A, we have:

Lemma 3.1.

Assume (F1)(F3). Then, for almost every λJ=[12,1], problem (3.1) possesses a bounded Palais–Smale sequence at the level cλ. Namely, there exists {un}H1(RN) such that

  • (i)

    {un} is bounded in H1(N),

  • (ii)

    Iλ(un)cλ and Iλ(un)0 in H-1(N) as n.

Next, in the spirit of [33, 41], we establish a decomposition of such a Palais–Smale sequence {un}, which will play a crucial role in proving the existence of ground states to (1.2). However, some extra difficulties with respect to the local case are carried over by the presence the nonlocal as well as critical H-L-S nonlinearity.

Proposition 3.1.

With the same assumptions in Theorem 1.2, let λ[12,1] and {un} given by Lemma 3.1. Assume unuλ weakly in H1(RN) as n. Then, up to a subsequence, there exist kN+, {xnj}j=1kRN and {vλj}j=1kH1(RN) such that:

  • (i)

    Iλ(uλ)=0 in H-1(N),

  • (ii)

    vλj0 and Iλ(vλj)=0 in H-1(N), j=1,2,,k,

  • (iii)

    cλ=Iλ(uλ)+j=1kIλ(vλj),

  • (iv)

    un-uλ-j=1kvλj(-xnj)0 as n,

  • (v)

    |xnj| and |xni-xnj| as n for any ij.

Before proving Proposition 3.1, we need a few preliminary lemmas.

Lemma 3.2.

Let λ[12,1] and let uλ be any nontrivial weak solution of (3.1). Then uλ satisfies the following Pohozǎev identity:

N-22N|uλ|2+N2aN|uλ|2=N+α2λN(IαF(uλ))F(uλ).(3.3)

Moreover, there exist β,γ>0 (independent of λ[12,1]) such that uλβ and Iλ(uλ)γ for any nontrivial solution uλ, λ[12,1].

Proof.

For the proof of the Pohozǎev-type identity (3.3) we refer to [45, Theorem 3]. Let λ[12,1] and let uλ be any nontrivial weak solution to (3.1). Then

N|uλ|2+a|uλ|2N(IαF(uλ))f(uλ)uλ.(3.4)

Thanks to (F1)(F2), for any ε>0, there exists Cε>0 such that F(t),tf(t)ε|t|N+αN+Cε|t|N+αN-2 for any t. Moreover, as in [45], there exists β>0 such that

N(IαF(u))f(u)uu22if uβ,

which yields by (3.4), uλβ. By Pohozǎev’s identity (3.3),

Iλ(uλ)=2+α2(N+α)N|uλ|2+αa2(N+α)N|uλ|2

and this concludes the proof. ∎

Let α(0,N). For any uD1,2(N), combining the Hardy–Littlewood–Sobolev inequality with Sobolev’s inequality, we have

N(Iα|u|N+αN-2)|u|N+αN-2Aα𝒞α(N|u|2NN-2)N+αNAα𝒞α𝒮-N+αN-2(N|u|2)N+αN-2,

where

𝒮:=inf0uD1,2(N)N|u|2(N|u|2NN-2)N-2N.

Then

𝒮α:=inf0uD1,2(N)N|u|2(N(Iα|u|N+αN-2)|u|N+αN-2)N-2N+α𝒮(Aα𝒞α)N-2N+α.

Minimizers for 𝒮α are explicitly known from [37, Theorem 4.3] (see also [27, Lemma 1.2]). Actually,

𝒮α=𝒮(Aα𝒞α)N-2N+α

and it is achieved by the instanton

U(x)=[N(N-2)]N-24(1+|x|2)N-22.

Now, we use this information to prove an upper estimate for cλ.

Lemma 3.3.

Let λ[12,1], α(0,N) and assume

q>max{1+αN-2,N+α2(N-2)}.

Then the following upper bound holds:

cλ<2+α2(N+α)(N+αN-2)N-22+αλ2-N2+α𝒮αN+α2+α.

Proof.

Let φC0(N) be a cut-off function with support B2 such that φ1 on B1 and 0φ1 on B2, where Br denotes the ball in N of center at origin and radius r. Given ε>0, we set ψε(x)=φ(x)Uε(x), where

Uε(x)=(N(N-2)ε2)N-24(ε2+|x|2)N-22.

By [8] (see also[61, Lemma 1.46]), we have the following estimates:

N|ψε|2=𝒮N2+{O(εN-2)if N4,K1ε+O(ε3)if N=3,N|ψε|2NN-2=𝒮N2+O(εN)if N3,N|ψε|2={K2ε2+O(εN-2)if N5,K2ε2|lnε|+O(ε2)if N=4,K2ε+O(ε2)if N=3,

where K1,K2>0. Then we get

N|ψε|2+a|ψε|2=𝒮N2+{aK2ε2+O(εN-2)if N5,aK2ε2|lnε|+O(ε2)if N=4,(K1+aK2)ε+O(ε2)if N=3.(3.5)

By direct computation, we know

(N|ψε|2NqN+α)N+αN=K3εN+α-(N-2)q+o(εN+α-(N-2)q),

and then by the Hardy–Littlewood–Sobolev inequality,

N(Iα|ψε|N+αN-2)|ψε|q𝒞α(N|ψε|2NN-2)N+α2N(N|ψε|2NqN+α)N+α2NK4εN+α-(N-2)q2+o(εN+α-(N-2)q2),(3.6)

where K3,K4>0. Moreover, similar as in [27, 25], by direct computation, for some K5>0,

N(Iα|ψε|N+αN-2)|ψε|N+αN-2(Aα𝒞α)N2𝒮αN+α2-K5εN+α2+o(εN+α2).(3.7)

We also have

N(Iα|ψε|N+αN-2)|ψε|qAα(NNUεN+αN-2(x)Uεq(y)|x-y|N-αdxdy-NB1B1UεN+αN-2(x)Uεq(y)|x-y|N-αdxdy-B1NB1UεN+αN-2(x)Uεq(y)|x-y|N-αdxdy-NB1NB1UεN+αN-2(x)Uεq(y)|x-y|N-αdxdy),

where for some K~i>0, i=1,2,3,4,

{NNUεN+αN-2(x)Uεq(y)|x-y|N-αdxdy=K~1εN+α-(N-2)q2,NB1B1UεN+αN-2(x)Uεq(y)|x-y|N-αdxdyK~2εN+α-N-22q+o(εN+α-N-22q),B1NB1UεN+αN-2(x)Uεq(y)|x-y|N-αdxdyK~3εN-22q+o(εN-22q),NB1NB1UεN+αN-2(x)Uεq(y)|x-y|N-αdxdyK~4εN+α+(N-2)q2+o(εN+α+(N-2)q2).

Thus for some K6>0, we have

N(Iα|ψε|N+αN-2)|ψε|qK6εN+α-(N-2)q2+o(εN+α-(N-2)q2).(3.8)

Here, we used the fact that q>N+α2(N-2). Then for any t>0,

Iλ(tψε)t22N|ψε|2+a|ψε|2-μλqN-2N+αtq+N+αN-2N(IαψεN+αN-2)ψεq-t2(N+α)N-22(N-2N+α)2λN(IαψεN+αN-2)ψεN+αN-2=:gε(t).

One has gε(t)- as t+ and gε(t)>0 for t>0 small. Following [55, Lemma 3.3], gε has a unique critical point tε in (0,+), which is the maximum point of gε. From gε(tε)=0,

tεN|ψε|2+a|ψε|2-(q+N+αN-2)μλqN-2N+αtεq+N+αN-2-1N(IαψεN+αN-2)ψεq=tε2(N+α)N-2-1N-2N+αλN(IαψεN+αN-2)ψεN+αN-2.(3.9)

Claim.

There exist t0,t1>0 (both independent of ε) such that tε[t0,t1] for ε>0 small.

Consider first the case, tε0 as ε0. Then by (3.5), (3.6) and (3.7), there exist c1,c2>0 (independent of ε) such that for ε small,

c1tεc2εN+α-(N-2)q2tεq+N+αN-2-1+tεq+N+αN-2-12tεq+N+αN-2-1,

where we used the fact that q<N+αN-2: hence a contradiction and tεt0. By (3.9), one has

N|ψε|2+a|ψε|2tε2(N+α)N-2-2N-2N+αλN(IαψεN+αN-2)ψεN+αN-2,

which implies, combining (3.5) and (3.7), that tεt1 for some t1>0 and ε small.

By the Claim just proved and (3.8), we have for some K7>0,

μλqN-2N+αtεq+N+αN-2N(IαψεN+αN-2)ψεqK7εN+α-(N-2)q2+o(εN+α-(N-2)q2)

and hence on the one hand the following:

maxt0Iλ(tψε)=gε(tε)tε22N|ψε|2+a|ψε|2-K7εN+α-(N-2)q2-tε2(N+α)N-22(N-2N+α)2λN(IαψεN+αN-2)ψεN+αN-2+o(εN+α-(N-2)q2)maxt0[t22N|ψε|2+a|ψε|2-t2(N+α)N-22(N-2N+α)2λN(IαψεN+αN-2)ψεN+αN-2]-K7εN+α-(N-2)q2+o(εN+α-(N-2)q2)=2+α2(N+α)(N+αN-2)N-22+αλ2-N2+α(N|ψε|2+a|ψε|2)N+α2+α(N(IαψεN+αN-2)ψεN+αN-2)N-22+α-K7εN+α-(N-2)q2+o(εN+α-(N-2)q2).

On the other hand, by (3.5) and (3.7), for some K8>0,

(N|ψε|2+a|ψε|2)N+α2+α(N(IαψεN+αN-2)ψεN+αN-2)N-22+α𝒮αN+α2+α+{K8εmin{2,N+α2}+o(εmin{2,N+α2})if N5,K8ε2|lnε|+o(ε2|lnε|)if N=4,K8ε+o(ε)if N=3.

Then, for some K9,K10>0,

maxt0Iλ(tψε)2+α2(N+α)(N+αN-2)N-22+αλ2-N2+α𝒮αN+α2+α+{K9εmin{2,N+α2}-K10εN+α-(N-2)q2+o(εN+α-(N-2)q2)if N5,K9ε2|lnε|-K10εN+α-(N-2)q2+o(εN+α-(N-2)q2)if N=4,K9ε-K10εN+α-(N-2)q2+o(εN+α-(N-2)q2)if N=3,<2+α2(N+α)(N+αN-2)N-22+αλ2-N2+α𝒮αN+α2+αif ε>0 is sufficiently small,

where we used the fact N+α-(N-2)q<min{2,N+α2}. Therefore, for any λ[12,1] and ε>0 small enough, we get

cλmaxt0Iλ(tψε)<2+α2(N+α)(N+αN-2)N-22+αλ2-N2+α𝒮αN+α2+α.

Proof of Proposition 3.1.

Let λ[12,1] and assume unuλ weakly in H1(N) and satisfy Iλ(un)cλ and Iλ(un)0 in H-1(N) as n.

Step 1. We claim Iλ(uλ)=0 in H-1(N). As a consequence of Lemma 2.4, it is enough to show, up to a subsequence, that for any fixed ϕC0(N),

N[IαF(un-u)]f(un-u)ϕ0as n.

In fact, by (F1)(F2), there exists C>0 such that

|f(t)|2NN+αC(|t|2NN+α+|t|2+αN-22NN+α),t.

By virtue of the Hardy–Littlewood–Sobolev inequality and Rellich’s theorem, up to a subsequence, for some C (independent of n) we have

|N[IαF(un-u)]f(un-u)ϕ|C(N|f(un-u)ϕ|2NN+α)N+α2N0as n.

Step 2. Set vn1:=un-uλ; we claim

limnsupzNB1(z)|vn1|2>0.(3.10)

Indeed, arguing by contradiction, if not, by Lions’ lemma [41, Lemma I.1], vn10 strongly in Lt(N) as n for any t(2,2NN-2). Noting that Iλ(un),vn10 as n and Iλ(uλ),vn1=0 for any n, by virtue of Lemma 2.2 and Lemma 2.4, we get

cλ=Iλ(uλ)+Iλ(vn1)+on(1),vn12=λN[IαF(vn1)]f(vn1)vn1+on(1),(3.11)

where on(1)0 as n. Next, we show that

limnN[IαF1(vn1)]F1(vn1)=0,

where

f1(t)=f(t)-|t|4+α-NN-2t,F1(t)=0tf1(s)ds,t.

Notice that 4NN+α(2,2NN-2) and f1(t)=o(t) as |t|0, lim|t||f1(t)||t|-2+αN-2=0. It is easy to see that

limnN|F1(vn1)|2NN+α=0,

which yields by the Hardy–Littlewood–Sobolev inequality that there exists some C>0 (independent of n) such that

|N[IαF1(vn1)]F1(vn1)|C(N|F1(vn1)|2NN+α)N+α2N0as n.

Similarly,

limnN[IαF1(vn1)]|vn1|N+αN-2=0,limnN[IαF1(vn1)]f1(vn1)vn1=0.

Then by (3.11), we get

{cλ=Iλ(uλ)+12vn12-λ2(N-2N+α)2N[Iα|vn1|N+αN-2]|vn1|N+αN-2+on(1),vn12=λN-2N+αN[Iα|vn1|N+αN-2]|vn1|N+αN-2+on(1),(3.12)

where on(1)0 as n. Recalling that vn10 strongly in H1(N) as n, let

limnvn12=λN-2N+αlimnN[Iα|vn1|N+αN-2]|vn1|N+αN-2=b;

then b>0. From

N|vn1|2𝒮α(N[Iα|vn1|N+αN-2]|vn1|N+αN-2)N-2N+α,

we have

b(N+αN-2)N-22+αλ2-N2+α𝒮αN+α2+α.

By Lemma 3.2 and (3.12),

cλ2+α2(N+α)(N+αN-2)N-22+αλ2-N2+α𝒮αN+α2+α,

which is a contradiction. Thus (3.10) holds true.

Step 3. By (3.10) and vn10 weakly in H1(N) as n, there exists {zn1}N such that |zn1| as n and

limnB1(zn1)|vn1|2>0.

Let un1=vn1(+zn1). Then, up to a subsequence, un1vλ1 weakly in H1(N) as n for some vλ10. By Lemma 2.2 and Lemma 2.4, we have

Iλ(un1)cλ-Iλ(uλ),Iλ(un1)0in H-1(N) as n.

Similarly as above, Iλ(vλ1)=0. Let vn2=un1-vλ1. Then

un=uλ+vλ1(-zn1)+vn2(-zn1).

If vn20, i.e. un1vλ1 strongly in H1(N) as n, then

cλ=Iλ(uλ)+Iλ(vλ1),un-uλ-vλ1(-zn1)0as n,

and we are done. Otherwise, if vn20 strongly in H1(N) as n, similarly as above

limnsupzNB1(z)|vn2|2>0.

Then there exists {zn2}N such that |zn2| as n and

limnB1(zn2)|vn2|2>0.

Let un2=vn2(+zn2). Then, up to a subsequence, un2vλ2 weakly in H1(N) as n for some vλ20. We have Iλ(vλ2)=0 and

Iλ(un2)cλ-Iλ(uλ)-Iλ(vλ1),Iλ(un2)0in H-1(N) as n.

Let vn3=un2-vλ2. Then

un=uλ+vλ1(-zn1)+vλ2(-zn1-zn2)+vn3(-zn1-zn2).

If vn30, i.e., un2vλ2 strongly in H1(N) as n, then

cλ=Iλ(uλ)+Iλ(vλ1)+Iλ(vλ2),un-uλ-vλ1(-zn1)-vλ2(-zn1-zn2)0as n,

and we are done. Otherwise, we can iterate the above procedure and by Lemma 3.2, we will end up in a finite number k of steps. Namely, let xnj=i=1jzni to have

cλ=Iλ(uλ)+j=1kIλ(vλj),un-uλ-j=1kvλj(-xnj)0as n.

Step 4. Clearly, |xnj-xnj-1|=|znj| as n for j=2,3,,k. However, it is not clear that if {xnj}j=1k repels each other as n, i.e., |xnj-xni| as n for any i,j=1,2,,k and ij. Let us show that after extracting a subsequence from {xnj} and redefining {vλj} if necessary, properties (iii), (iv), (v) hold. Let Λ1,Λ2{1,2,,k} and satisfy Λ1Λ2={1,2,,k} and let {xnj}n be bounded if jΛ1, whereas |xnj| as n if jΛ2. Then, for any jΛ1 if Λ1, there exists 0vjH1(N) such that, up to a subsequence, vλj(-xnj)vj weakly in H1(N) as n and Iλ(vj)=0 in H-1(N). By Rellich’s theorem, for any t[2,2NN-2), we have vλj(-xnj)vj strongly in Lt(N) as n. Noting that Iλ(vλj(-xnj))=0 in H-1(N) and Iλ(vλj(-xnj))cλ, similar to Step 2, we know that vλj(-xnj)vj strongly in H1(N) as n. Then, up to a subsequence, there exists v~jH1(N) such that jΛ1vλj(-xnj)v~j strongly in H1(N) as n, which eventually implies

un-uλ-jΛ2vλj(-xnj)0as n.

Recalling that un-uλ0 as n, we have Λ2. Let xniΛ2 and

Λ2i:={jΛ2:|xni-xnj| stays bounded}.

Then similarly as above, up to a subsequence, for some v~λiH1(N), we have jΛ2ivλj(+xni-xnj)v~λi strongly in H1(N) as n. Then, as n,

un-uλ-v~λi(-xni)-j(Λ2Λ2i)vλj(-xnj)0.

Without loss of generality, we may assume that v~λi0. Noting that un(+xni)v~λi a.e. in N as n, we get Iλ(v~λi)=0 in H-1(N). Then we redefine vλi:=v~λi and as n,

un-uλ-j(Λ2Λ2i){i}vλj(-xnj)0.

By repeating the argument above at most (k-1) times and redefining {vλj} if necessary, we end up with ΛΛ2 such that

{|xnj|and|xni-xnj|as n for any i,jΛ and ij,un-uλ-jΛvλj(-xnj)0as n.

Finally, by Lemma 2.2 one has cλ=Iλ(uλ)+jΛIλ(vλj). The proof is now complete. ∎

Proof of Theorem 1.1.

As a consequence of Lemma 3.1, Proposition 3.1 and Lemma 3.2, one has that for almost every λJ=[12,1], problem (3.1) admits a nontrivial solution uλ satisfying uλβ, γIλ(uλ)cλ, where β,γ>0 (independent of λ). Then there exist {λn}[12,1] and {un}H1(N) such that, as n,

λn1,γIλn(un)cλn,Iλn(un)=0in H-1(N).(3.13)

By Pohozǎev’s identity (3.3) we have

Iλn(un)=2+α2(N+α)N|un|2+αa2(N+α)N|un|2

and {un} is bounded in H1(N). Notice that

La(u)=Iλ(u)+12(λ-1)N(IαF(u))F(u),uH1(N).

Then by (3.13), up to a sequence, there exists c0[γ,c1] such that

c0:=limnLa(un)=limnIλn(un)limncλn=c1,

where we used the fact that cλ is continuous from the left at λ. Moreover, by (3.13), for any ϕC0(N),

La(un),ϕ=(λn-1)N[IαF(un)]f(un)ϕ.

Similarly as above, there exists some C>0 such that

(N|f(un)ϕ|2NN+α)N+α2NCϕuniformly for all ϕC0(N),n=1,2,,

and by the Hardy–Littlewood–Sobolev inequality

|La(un),ϕ|=(1-λn)|N[IαF(un)]f(un)ϕ|C(1-λn)(N|F(un)|2NN+α)N+α2N(N|f(un)ϕ|2NN+α)N+α2N=on(1)ϕ,

where on(1)0 uniformly for any ϕC0(N) as n. Namely, La(un)0 in H-1(N) as n. Finally, we obtain

unβ,La(un)c0c1,La(un)0in H-1(N) as n.

If unu0 strongly in H1(N), then u0β, La(u0)=c0c1 and La(u0)=0 in H-1(N). Otherwise, as a consequence of Proposition 3.1 with λ=1,cλ=c0,uλ=u0, there exist k+ and {vj}j=1kH1(N) such that vj0, La(vj)=0 in H-1(N) for all j and c0=La(u0)+j=1kLa(vj). So let

𝒩:={uH1(N{0}):La(u)=0in H-1(N)}.

Then 𝒩 and infu𝒩La(u)=Ea[γ,c1].

We conclude the proof of Theorem 1.1 by showing that Ea is achieved. Clearly, there exists {vn}𝒩 such that as n, La(vn)Ea and La(vn)=0 in H-1(N). Thus {vn} is bounded in H1(N). Assume that vnv0 weakly in H1(N) as n. Then La(v0)=0 in H-1(N). If vnv0 strongly in H1(N), then La(v0)=Ea. Namely, v0 is a ground state solution of (1.2). Otherwise, there exist k+ and {vj}j=1kH1(N) such that vj0, La(vj)=0 in H-1(N) for all j and Ea=La(v0)+j=1kLa(vj). By the definition of Ea, v0=0, k=1 and La(v1)=Ea, which yields v1 as a ground state solution to (1.2). The proof is now complete. ∎

4 Towards semiclassical states

4.1 Compactness of the set of ground state solutions

Denote the set of ground state solutions to (1.2) by

𝒩a:={uH1(N):La(u)=Ea,La(u)=0 in H-1(N)}.

Then by Theorem 1.1, 𝒩a for any a>0. Since La is invariant by translations, 𝒩a cannot be compact in H1(N). However, this turns out to be the only way to loose compactness as we have the following result.

Proposition 4.1.

For any a>0, up to translations, Na is compact in H1(RN).

Proof.

Let {un}𝒩a. Then La(un)=Ea and La(un)=0 in H-1(N). Similarly as above {un} is bounded in H1(N). Assume that unu0 weakly in H1(N) as n; then La(u0)=0 in H-1(N). If unu0 strongly in H1(N), we are done. Otherwise, by virtue of Proposition 3.1, up to a subsequence, there exists k+, {xnj}j=1kN and {vj}j=1kH1(N) such that vj0, La(vj)=0 in H-1(N) for all j and

Ea=La(u0)+j=1kLa(vj),un-u0-j=1kvλj(-xnj)0as n,

which implies that u0=0, k=1, v1𝒩a and un(+xn1)-vλ10 as n. ∎

4.2 Regularity, positivity and symmetry

Here we borrow some ideas from [45, 4] to establish boundedness, decay, positivity and symmetry of ground state solutions to (1.2).

Proposition 4.2.

Let a>0. The following hold:

  • (i)

    0<inf{u:u𝒩a}sup{u:u𝒩a}<.

  • (ii)

    For any u𝒩a, uCloc1,γ(N) for γ(0,1).

  • (iii)

    For any u𝒩a, u has constant sign and is radially symmetric about a point.

  • (iv)

    Ea coincides with the mountain pass value.

  • (v)

    There exist C,c>0 , independent of u𝒩a , such that |Dα1u(x)|Cexp(-c|x-x0|), xN , for |α1|=0,1 , where |u(x0)|=maxxN|u(x)|.

Proof.

First, by Pohozaev’s inequality it follows that 𝒩a is bounded in H1(N).

Claim 1.

For any p[2,Nα2NN-2), there exists Cp>0 such that

upCpu2for all u𝒩a.(4.1)

In fact, for any fixed u𝒩a, let H(u)=F(u)u and K(u)=f(u) in {xN:u(x)0}. Let R>0 and ϕRC0() be such that ϕR(t)[0,1] for t, ϕR(t)=1 for |t|R and ϕR(t)=0 for |t|2R. Set

H(u)=ϕR(u)H(u),H(u)=H(u)-H(u),K(u)=ϕR(u)K(u),K(u)=K(u)-K(u).

By (F1)(F2), there exists C>0 (depending only on R) such that for any xN,

|H(u)|C|u|αN,|K(u)|C|u|αN,|H(u)|C|u|α+2N-2,|K(u)|C|u|α+2N-2.

Note that H(u),K(u) are uniformly bounded in L2N/α(N) and so are H(u),K(u) in L2N/(α+2)(N) for any u𝒩a. Thanks to the compactness of 𝒩a, for any ε>0 we can choose R depending only on ε such that

(N|H(u)|2Nα+2N|K(u)|2Nα+2)α+22Nε2for all u𝒩a.

Then repeating line by line the argument as in [45, Proposition 3.1], (4.1) follows.

Claim 2.

The map IαF(u) is uniformly bounded in L(RN) for all uNa.

By (F1)(F2) and the very definition of IαF(u), there exists C(α) (depending only N,α) such that for any xN and u𝒩a,

(Iα|F(u)|)(x)C(α)2(|u|2+|u|N+αN-2)dy+C(α)|x-y|1|u|2+|u|N+αN-2|x-y|N-αdy.

Thanks to (4.1), for some c (independent of u) such that for any xN,

(Iα|F(u)|)(x)c+C(α)|x-y|1|u|2+|u|N+αN-2|x-y|N-αdy.

As in [64, Proposition 2.2], we can choose t(Nα,NαNN-2) with 2t(2,Nα2NN-2) and s(Nα,Nα2NN+α) with sN+αN-2(2,Nα2NN-2), and there exist C1,C2>0 (independent of u) such that

|x-y|1|u|2+|u|N+αN-2|x-y|N-αdyC1u2t2+C2usN+αN-2N+αN-2,

which combining with (4.1) implies the claim.

Now let f¯(x,u):=(IαF(u))(x)f(u). Then by (F1)(F2), for any u𝒩a, u satisfies that for any δ>0, there exists Cδ>0 (independent of u) such that

|f¯(x,u)u|(δ|u|2+Cδ|u|N+αN-2),xN,

and

-Δu+au=f¯(x,u),uH1(N).

Noting that N+αN-2<2NN-2, by means of a standard Moser iteration [29] (see also [14]), 𝒩a is uniformly bounded in L(N). Since |f¯(x,u)|=o(1)|u| if u0 and Ea>0, one also has inf{u:u𝒩a}>0.

Since uL(N) for any 𝒩a, it follows from the elliptic regularity estimates (see [29]) that uCloc1,γ(N) for some γ(0,1). From the proof of Theorem 1.1, we know that Eac1, where

c1:=infγΓmaxt[0,1]La(γ(t)),

where Γ:={γC([0,1],X):γ(0)=0,La(γ(1))<0}. Following [45], for any u𝒩a, there exists a path γΓ such that γ(12)=u and La(γ) achieves its maximum at 12. Thereby, c1=Ea. Namely, Ea is also a mountain pass value. Moreover, for any u𝒩a, u has a constant sign and is radially symmetric about some point. If u is positive, then u is decreasing at r=|x-x0|, where x0 is the maximum point of u. Finally, by the radial lemma, u(x)0 uniformly as |x-x0| for u𝒩a. By the comparison principle, there exist C,c>0, independent of u𝒩a such that |Dα1u(x)|Cexp(-c|x-x0|),xN for |α1|=0,1. ∎

4.3 Proof of Theorem 1.2

Let u(x)=v(εx), Vε(x)=V(εx) and consider the following problem:

-Δu+Vε(x)u=(IαF(u))f(u),xN.(4.2)

Let Hε be the completion of C0(N) with respect to the norm

uε=(N(|u|2+Vεu2))12.

For any set BN and ε>0, we define Bε{xN:εxB} and Bδ{xN:dist(x,B)δ}. Since we are looking for positive solutions of (1.1), from now on, we may assume that f(t)=0 for t0. For uHε, let

Pε(u)=12N|u|2+Vεu2-12N(IαF(u))F(u).

Fix an arbitrary ν>0 and define

χε(x)={0if xOε,ε-νif xNOε,

as well as

Qε(u)=(Nχεu2dx-1)+2.

Let Γε:Hε be given by

Γε(u)=Pε(u)+Qε(u).

To find solutions of (4.2) which concentrate inside O as ε0, we look for critical points uε of Γε satisfying Qε(uε)=0. The functional Qε that was first introduced in [13] will act as a penalization to forcing the concentration phenomena inside O. In what follows, we seek the critical points of Γε in some neighborhood of ground state solutions to (1.2) with a=m.

4.4 The truncated problem

Denote Sm by the set of positive ground state solutions of (1.2) with a=m satisfying u(0)=maxxNu(x), where m is given in Section 1.

Lemma 4.1.

The set Sm is compact in H1(RN).

Proof.

By Proposition 4.2, Sm. For any {un}Sm, without loss of generality, we assume that unu0 weakly in H1(N) and a.e. in N as n. Let us first prove that u00. Indeed, by (v) of Proposition 4.2, there exist c,C>0 (independent of n) such that |un(x)|Cexp(-c|x|) for any xN. By the Lebesgue dominated convergence theorem, unu0 strongly in Lp(N) as n for any p[2,2NN-2]. So if u0=0, one has un0 strongly in H1(N) as n, which contradicts the fact Em>0. We claim unu0 strongly in H1(N) as n. Indeed, if not, by Proposition 3.1, there exist k+ and {vj}j=1kH1(N) such that vj0, Lm(vj)=0 in H-1(N) for all j and Em=Lm(u0)+j=1kLm(vj). Noting that Lm(u0)Em and Lm(vj)Em, we get a contradiction. Finally, u0Sm. Clearly, u0𝒩m is positive and radially symmetric. Recalling that 0 is the same maximum point un for any n, by the local elliptic estimate, 0 is also a maximum point of u0. The proof is complete. ∎

By Proposition 4.2, let κ>0 be fixed and satisfy

supUSmU<κ.(4.3)

For k>maxt[0,κ]f(t) fixed, let fk(t):=min{f(t),k} and consider the truncated problem

-ε2Δv+V(x)v=ε-α(IαFk(v))fk(v),vH1(N),(4.4)

whose associated limit problem is

-Δu+mu=(IαFk(u))fk(u),uH1(N),(4.5)

where Fk(t)=0tfk(s)ds. Denote by Smk be the set of positive ground state solutions U of (4.5) satisfying U(0)=maxxNU(x). Then by [45, Theorem 2], Smk. As in Lemma 4.1, Smk is compact in H1(N).

Lemma 4.2.

We have SmSmk.

Proof.

Denote by Emk the least energy of (4.5). Notice that any uSm is also a solution to (4.5). Then EmkEm. By [45], Emk is a mountain pass value. Combining (iv) of Proposition 4.2 with the fact fk(t)f(t) for t>0 and fk(t)=f(t)=0 for t0, we have EmkEm and so Emk=Em, which yields SmSmk. ∎

4.5 Proof of Theorem 1.2

In the following, we use the truncation approach to prove Theorem 1.2. First, we consider the truncated problem (4.4). By Lemma 4.2, Sm is a compact subset of Smk. Inspired from [10] we show that (4.4) admits a nontrivial positive solution vε in some neighborhood of Sm for small ε. Then we show that there exists ε0>0 such that

vε<κfor ε(0,ε0).

As a consequence, vε turns out to be a solution to the original problem (1.1).

For this purpose, set

δ=110min{dist(,Oc)}.

Let β(0,δ) and consider a cut-off φC0(N) such that 0φ1, φ(x)=1 for |x|β and φ(x)=0 for |x|2β. Set φε(y)=φ(εy), yN, and for some x()β and USm, we define

Uεx(y)=φε(y-xε)U(y-xε)

and

Xε={Uεx:x()β,UiSm}.

In the following, we show that (4.4) admits a solution in XεdXε for ε,d>0 small enough, where

Xεd={uHε:infvXεu-vεd}.

In fact, since fk satisfies all the hypotheses of [64, Theorem 2.1], for ε,d>0 small, (4.4) admits a positive solution vεXεd for which there exist USm and a maximum point xε of vε such that limε0dist(xε,)=0 and vε(ε+xε)U(+z0) in H1(N) as ε0 for some z0N. We have

-Δwε+Vε(x+xεε)wε=(IαFk(wε))fk(wε),xN,

where wε()=vε(ε+xε). As in Proposition 4.2, IαFk(wε) is uniformly bounded in L(N) for all ε. Then, noting that 0fk(wε(x))k for all xN, local elliptic estimates (see [29]) yield wε(0)U(z0) as ε0. It follows from (4.3) that vε=wε(0)<κ uniformly for small ε>0. Therefore, for small ε>0, fk(vε(x))f(vε(x)), xN, and then vε is a positive solution to (1.1). ∎

Conflict of interest.

The authors declare they have no conflict of interest.

References

  • [1]

    N. Ackermann and T. Weth, Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting, Commun. Contemp. Math. 7 (2005), no. 3, 269–298.  CrossrefGoogle Scholar

  • [2]

    C. O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in 2, J. Differential Equations 261 (2016), no. 3, 1933–1972.  Google Scholar

  • [3]

    C. O. Alves, J. A. M. do Ó and M. A. S. Souto, Local mountain-pass for a class of elliptic problems in N involving critical growth, Nonlinear Anal. 46 (2001), no. 4, 495–510.  Google Scholar

  • [4]

    C. O. Alves, F. Gao, M. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations 263 (2017), no. 7, 3943–3988.  CrossrefGoogle Scholar

  • [5]

    C. O. Alves, M. Squassina and M. Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 23–58.  CrossrefGoogle Scholar

  • [6]

    H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313–345.  CrossrefGoogle Scholar

  • [7]

    C. Bonanno, P. d’Avenia, M. Ghimenti and M. Squassina, Soliton dynamics for the generalized Choquard equation, J. Math. Anal. Appl. 417 (2014), no. 1, 180–199.  CrossrefGoogle Scholar

  • [8]

    H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477.  CrossrefGoogle Scholar

  • [9]

    J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity, Trans. Amer. Math. Soc. 362 (2010), no. 4, 1981–2001.  Google Scholar

  • [10]

    J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal. 185 (2007), no. 2, 185–200.  CrossrefGoogle Scholar

  • [11]

    J. Byeon and K. Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1859–1899.  CrossrefGoogle Scholar

  • [12]

    J. Byeon and K. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Mem. Amer. Math. Soc. 229 (2014), no. 1076, 1–89.  Google Scholar

  • [13]

    J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. II, Calc. Var. Partial Differential Equations 18 (2003), no. 2, 207–219.  CrossrefGoogle Scholar

  • [14]

    J. Byeon, J. Zhang and W. Zou, Singularly perturbed nonlinear Dirichlet problems involving critical growth, Calc. Var. Partial Differential Equations 47 (2013), no. 1–2, 65–85.  CrossrefGoogle Scholar

  • [15]

    W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), no. 3, 330–343.  CrossrefGoogle Scholar

  • [16]

    S. Cingolani, S. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 5, 973–1009.  CrossrefGoogle Scholar

  • [17]

    P. d’Avenia, A. Pomponio and D. Ruiz, Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods, J. Funct. Anal. 262 (2012), no. 10, 4600–4633.  CrossrefGoogle Scholar

  • [18]

    M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), no. 2, 121–137.  CrossrefGoogle Scholar

  • [19]

    M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), no. 2, 127–149.  CrossrefGoogle Scholar

  • [20]

    M. del Pino and P. L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (1999), no. 3, 883–898.  Google Scholar

  • [21]

    M. D. Donsker and S. R. S. Varadhan, The polaron problem and large deviations, Phys. Rep. 77 (1981), no. 3, 235–237.  CrossrefGoogle Scholar

  • [22]

    M. D. Donsker and S. R. S. Varadhan, Asymptotics for the polaron, Comm. Pure Appl. Math. 36 (1983), no. 4, 505–528.  CrossrefGoogle Scholar

  • [23]

    A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), no. 3, 397–408.  CrossrefGoogle Scholar

  • [24]

    H. Fröhlich, Theory of electrical breakdown in ionic crystal, Proc. Roy. Soc. Ser. A 160 (1937), 230–241.  CrossrefGoogle Scholar

  • [25]

    F. Gao and M. Yang, Existence and multiplicity of solutions for a class of Choquard equations with Hardy–Littlewood–Sobolev critical exponent, preprint (2016), https://arxiv.org/abs/1605.05038v1.  

  • [26]

    F. Gao and M. Yang, On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents, J. Math. Anal. Appl. 448 (2017), no. 2, 1006–1041.  CrossrefGoogle Scholar

  • [27]

    F. Gao and M. Yang, On the Brezis–Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math. (2017), 10.1007/s11425-000-0000-0.  Google Scholar

  • [28]

    B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in 𝐑n, Mathematical Analysis and Applications. Part A, Adv. in Math. Suppl. Stud. 7, Academic Press, New York (1981), 369–402.  Google Scholar

  • [29]

    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1983.  Google Scholar

  • [30]

    E. P. Gross, Physics of many-Particle Systems. Vol. 1, Gordon Breach, New York, 1996.  Google Scholar

  • [31]

    L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28 (1997), no. 10, 1633–1659.  CrossrefGoogle Scholar

  • [32]

    L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on 𝐑N, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 787–809.  Google Scholar

  • [33]

    L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on N, Indiana Univ. Math. J. 54 (2005), no. 2, 443–464.  Google Scholar

  • [34]

    L. Jeanjean and J. F. Toland, Bounded Palais–Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 1, 23–28.  CrossrefGoogle Scholar

  • [35]

    E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1976/77), no. 2, 93–105.  Google Scholar

  • [36]

    E. H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374.  CrossrefGoogle Scholar

  • [37]

    E. H. Lieb and M. Loss, Analysis, 2nd ed., Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001.  Google Scholar

  • [38]

    E. H. Lieb and B. Simon, The Hartree–Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977), no. 3, 185–194.  CrossrefGoogle Scholar

  • [39]

    P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal. 4 (1980), no. 6, 1063–1072.  CrossrefGoogle Scholar

  • [40]

    P.-L. Lions, Compactness and topological methods for some nonlinear variational problems of mathematical physics, Nonlinear Problems: Present and Future (Los Alamos 1981), North-Holland Math. Stud. 61, North-Holland, Amsterdam, (1982), 17–34.  Google Scholar

  • [41]

    P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223–283.  CrossrefGoogle Scholar

  • [42]

    L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), no. 2, 455–467.  CrossrefGoogle Scholar

  • [43]

    M. Macrì and M. Nolasco, Stationary solutions for the non-linear Hartree equation with a slowly varying potential, NoDEA Nonlinear Differential Equations Appl. 16 (2009), no. 6, 681–715.  CrossrefGoogle Scholar

  • [44]

    V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), no. 2, 153–184.  CrossrefGoogle Scholar

  • [45]

    V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579.  Google Scholar

  • [46]

    V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations 52 (2015), no. 1–2, 199–235.  CrossrefGoogle Scholar

  • [47]

    V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl. 19 (2017), no. 1, 773–813.  CrossrefGoogle Scholar

  • [48]

    M. Nolasco, Breathing modes for the Schrödinger–Poisson system with a multiple-well external potential, Commun. Pure Appl. Anal. 9 (2010), no. 5, 1411–1419.  CrossrefGoogle Scholar

  • [49]

    Y.-G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V)a, Comm. Partial Differential Equations 13 (1988), no. 12, 1499–1519.  Google Scholar

  • [50]

    S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie, Berlin, 1954.  Google Scholar

  • [51]

    R. Penrose, On gravity’s role in quantum state reduction, Gen. Relativity Gravitation 28 (1996), no. 5, 581–600.  CrossrefGoogle Scholar

  • [52]

    R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356 (1998), no. 1743, 1927–1939.  CrossrefGoogle Scholar

  • [53]

    R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf, New York, 2005.  Google Scholar

  • [54]

    P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), no. 2, 270–291.  CrossrefGoogle Scholar

  • [55]

    D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), no. 2, 655–674.  CrossrefGoogle Scholar

  • [56]

    S. Secchi, A note on Schrödinger–Newton systems with decaying electric potential, Nonlinear Anal. 72 (2010), no. 9–10, 3842–3856.  CrossrefGoogle Scholar

  • [57]

    M. Struwe, The existence of surfaces of constant mean curvature with free boundaries, Acta Math. 160 (1988), no. 1–2, 19–64.  CrossrefGoogle Scholar

  • [58]

    X. Sun and Y. Zhang, Multi-peak solution for nonlinear magnetic Choquard type equation, J. Math. Phys. 55 (2014), no. 3, Article ID 031508.  Google Scholar

  • [59]

    X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys. 153 (1993), no. 2, 229–244.  CrossrefGoogle Scholar

  • [60]

    J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger–Newton equations, J. Math. Phys. 50 (2009), no. 1, Article ID 012905.  Google Scholar

  • [61]

    M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996.  Google Scholar

  • [62]

    M. Willem, Functional Analysis. Fundamentals and Applications, Cornerstones, Birkhäuser, New York, 2013.  Google Scholar

  • [63]

    M. Yang and Y. Ding, Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part, Commun. Pure Appl. Anal. 12 (2013), no. 2, 771–783.  Google Scholar

  • [64]

    M. Yang, J. Zhang and Y. Zhang, Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity, Commun. Pure Appl. Anal. 16 (2017), no. 2, 493–512.  CrossrefGoogle Scholar

  • [65]

    J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, J. Lond. Math. Soc. (2) 90 (2014), no. 3, 827–844.  CrossrefGoogle Scholar

  • [66]

    J. Zhang, J. A. M. do Ó and M. Squassina, Schrödinger–Poisson systems with a general critical nonlinearity, Commun. Contemp. Math. 19 (2017), no. 4, Article ID 1650028.  Google Scholar

  • [67]

    J. Zhang and W. Zou, A Berestycki–Lions theorem revisited, Commun. Contemp. Math. 14 (2012), no. 5, Article ID 1250033.  Google Scholar

  • [68]

    J. Zhang and W. Zou, Solutions concentrating around the saddle points of the potential for critical Schrödinger equations, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 4119–4142.  CrossrefGoogle Scholar

About the article

Received: 2018-01-23

Accepted: 2018-03-14

Published Online: 2018-06-07


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1184–1212, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0019.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[2]
Wangcheng Huang, Wei Long, Aliang Xia, and Xiongjun Zheng
Boundary Value Problems, 2019, Volume 2019, Number 1

Comments (0)

Please log in or register to comment.
Log in