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 …

Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent

Mingqi Xiang / Binlin Zhang
  • Corresponding author
  • College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, 266590, P.R. China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Vicenţiu D. Rădulescu
  • Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland and Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2019-08-06 | DOI: https://doi.org/10.1515/anona-2020-0021

Abstract

This paper concerns the existence and multiplicity of solutions for the Schrődinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent. As a particular case, we study the following degenerate Kirchhoff-type nonlocal problem:

uλ(θ1)p[λ(Δ)psu+V(x)|u|p2u]=|u|ps2u+f(x,u)inN,uλ=(λ2N|u(x)u(y)|p|xy|N+psdxdy+NV(x)|u|pdx)1/p

where (Δ)psis the fractional p–Laplacian with 0 < s < 1 < p < N/s, ps=Np/(Nps)is the critical fractional Sobolev exponent, λ > 0 is a real parameter, 1<θps/p,and f : ℝN × ℝ ℝ is a Carathéodory function satisfying superlinear growth conditions. For θ(1,ps/p),by using the concentration compactness principle in fractional Sobolev spaces, we show that if f(x, t) is odd with respect to t, for any m ∈ ℕ+ there exists a Λm > 0 such that the above problem has m pairs of solutions for all λ ∈ (0, Λm]. For θ=ps/p,by using Krasnoselskii’s genus theory, we get the existence of infinitely many solutions for the above problem for λ large enough. The main features, as well as the main difficulties, of this paper are the facts that the Kirchhoff function is zero at zero and the potential function satisfies the critical frequency infx∈ℝ V(x) = 0. In particular, we also consider that the Kirchhoff term satisfies the critical assumption and the nonlinear term satisfies critical and superlinear growth conditions. To the best of our knowledge, our results are new even in p–Laplacian case.

Keywords: Schrödinger–Kirchhoff problem; Fractional p–Laplacian; Multiple solutions; Critical exponent; Principle of concentration compactness

MSC 2010: 35R11; 35A15; 47G20

1 Introduction

In this article we concern with existence and multiplicity of solutions for critical Kirchhoff–type problems involving the fractional p–Laplacian. More precisely, we consider

M(uλp)[λ(Δ)psu+V(x)|u|p2u]=|u|ps2u+f(x,u)inN,uλ=(λ2N|u(x)u(y)|p|xy|N+psdxdy+NV(x)|u|pdx)1/p,(1.1)

where ps=Np/(Nsp), N > sp with s ∈ (0, 1), (Δ)psis the fractional p–Laplacian which (up to normalization factors) may be defined for any x ∈ ℝN as

(Δ)psφ(x)=2limδ0N\Bδ(x)|φ(x)φ(y)|p2(φ(x)φ(y))|xy|N+psdy

for any φC0(N),where Bδ(x) denotes the ball in ℝN centered at x with radius δ. For a simple introduction about the fractional p–Laplacian, we refer to [1] and the references therein.

Furthermore, we always assume M, V and f satisfy the following assumptions:

(M) MC(ℝ, ℝ) and there exist θ(1,ps/p)and 0 < m0m1 such that

m0tθ1M(t)m1tθ1forallt0+;

(V) VC(ℝN, ℝ), V(x0)=minxNV(x) = 0 and there exists a constant h > 0 such that the Lebesgue measure of set Vh = {x ∈ ℝN : V(x) < h} is finite; there is ϱ > 0 such that lim|y|meas ({x ∈ Bϱ(y) : V(x) < c}) = 0 for any c ∈ ℝ+;

(f1) f : ℝN × ℝ ℝ is a Carathéodory function and there exists q(θp,ps)such that for any ε > 0 there exists Cε > 0 and

|f(x,ξ)|θpε|ξ|θp1+qCε|ξ|q1fora.exNandallξ;

(f2) There exists q1 > m1θp/m0 such that

q1F(x,ξ)f(x,ξ)ξforall(x,ξ)N×,

where F(x,ξ)=0ξf(x,τ)dτ,m0 and m1 are the numbers given in (M);

(f3) There exists q2(θp,ps)such that F(x, ξ) ≥ a0|ξ|q2 for a.e. x ∈ ℝN and all ξ ∈ ℝ.

Note that condition (V2), which is weaker than the coercivity assumption: V(x) ∞ as |x| ∞, was first introduced by Bartsch and Wang in [2] to conquer the lack of compactness.

In the last few years, great attention has been paid to the study of non-local fractional Laplacian problems involving critical nonlinearities. It is worth mentioning that the semilinear Laplace equation of elliptic type involving critical exponent was investigated in the crucial paper of Brézis and Nirenberg [3]. After that, many researchers dedicated to the study of several kinds of elliptic equations with critical growth in bounded domains or in the whole space. For example, by variational techniques, Servadei and Valdinoci [4] showed a Brézis–Nirenberg type result for non-local fractional Laplacian in bounded domains with homogeneous Dirichlet boundary datum, see also [5] for further discussions. In [6] Ros-Oton and Serra considered nonexistence results for nonlocal equations involving critical and supercritical nonlinearities. Autuori and Pucci [7] obtained a multiplicity result for fractional Laplacian problems in ℝN by using the mountain pass theorem and the direct method in variational methods, in which one of two superlinear nonlinearities could be critical or even supercritical.

Indeed, the interest in the study of partial differential equations involving the non-local fractional Laplacian goes beyond the mathematical curiosity. This type of non-local operator comes to real world with many different applications in a quite natural way, such as finance, ultra-relativistic limits of quantum mechanics, materials science, water waves, phase transition phenomena, anomalous diffusion, soft thin films, minimal surfaces and game theory, see for example [1, 8, 9] and the references therein. The literature on fractional Laplace operators and their applications is quite large and interesting, here we just list a few, see [10, 11, 12] and the references therein. For the basic properties of fractional Sobolev spaces and the study of fractional Laplacian based on variational methods, we refer the readers to [1, 13]. It is worth pointing out that one of the reasons that forced the rapid expansion of the fractional Laplacian results has been the nonlinear fractional Schrödinger equation, which was proposed by Laskin [14, 15] as a result of expanding the Feynman path integral, from the Brownian–like to the Lévy–like quantum mechanical paths.

In the last decade, the existence and multiplicity of solutions for the Kirchhoff–type elliptic equations with critical exponents have attracted much interest of many scholars. For instance, we refer to [16, 17, 18] for the setting of bounded domains; we collect also some articles, see [19, 20, 21] for the context set in the whole space. In particular, Fiscella and Valdinoci [22] proposed a stationary Kirchhoff–type equation which models the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. More precisely, they considered a model as follows:

{M(2N|u(x)u(y)|2|xy|N+2sdxdy)(Δ)su=λf(x,u)+|u|2s2uinΩu=0inN\Ω,(1.2)

where M(y) = α + βy for all y ≥ 0, here α > 0, β ≥ 0. Note that M is this type, problem (1.2) is called nondegenerate if α > 0 and β ≥ 0, while it is named degenerate if α = 0 and β > 0, see [23] for some physical motivation about degenerate Kirchhoff problems. For more details about the physical background of the fractional Kirchhoff model,we refer to [22, Appendix A]. Afterwards, the fractional Kirchhoff–type problems have been extensively investigated, for example, we refer to [24, 25, 26, 27] for some recent results about non-degenerate Kirchhoff–type problems.

In the following, let us recall some existence results about degenerate Kirchhoff–type fractional p– Laplacian problems. By using the mountain pass theorem and Ekeland’s variational principle, Xiang et al. [28] obtained the existence of two solutions for a nonhomogeneous Kirchhoff type problem driven by the fractional p–Laplacian, where the nonlinearity is convex-concave, see [26] for related results obtained by the same methods. In [29], Mingqi et al. investigated the existence of infinitely many solutions for Kirchhoff type fractional p–Laplacian problems, in which the symmetric mountain pass theorem is applied to study the suplinear case and the Krasnoselskii’s genus theory is used to consider the sublinear case. In [23], Pucci et al. studied the existence and multiplicity of entire solutions for a class of fractional p–Laplacian problems of Kirchhoff type via variational methods and topological degree theory. In [30], Mingqi et al. considered the multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p–Laplacian by using the Nehari manifold method and the symmetric mountain pass theorem. Evidently, the above works did not involve the critical case. For the critical case, with the help of Kajikiya’s new version of the symmetric mountain pass lemma, the existence of infinitely many solutions for a critical problem similar to (1.1) is proved in [31], see [32, 33] for more related results.

However, there are few results in the available literature on problems like problem (1.1). In particular, there are no result on the multiplicity of solutions for problem (1.1). There is no doubt that we encounter serious difficulties because of the lack of compactness and of the nonlocal nature of the fractional p–Laplacian. To overcome the loss of compactness, Xiang et al. [34] extended the concentration compactness principle of Lions [35] to the setting of fractional p–Laplacian in ℝN, and used it to get the existence of solutions for the following critical p–Kirchhoff problem

[a+b(2N|u(x)u(y)|p|xy|N+psdxdy)θ1](Δ)psu(x)=|u(x)|ps2u(x)+λf(x)inN,

where a ≥ 0, b > 0, θ > 1, λ > 0 is a parameter and fLpsps1(N).In [36], Fiscella and Pucci studied the following p–Kirchhoff problem involving critical Hardy-Sobolev nonlinearity

M(up)[(Δ)psu+V(x)|u|p2u]|u|ps(α)2u|x|α=λf(x,u)+g(x,u)inN,

where ps(α)=(Nα)pNpsis the critical Hardy-Sobolev exponent with α ∈ [0, ps), f and g are subcritical nonlinear terms, and VC(ℝN , ℝ) with infxNV(x)V0>0.Moreover, the existence of infinitely many solutions for problem (1.1) is investigated, assuming infxNV(x)=0and the subcritical nonlinear term f satisfies superlinear growth condition. In [37], Byeon and Wang first studied the asymptotic behavior of positive solutions to Schrodinger equations under the condition infxNV(x)=0,which is called critical frequency. In [38], Cao and Noussair extended the results of Byeon and Wang,and studied multi-bump standing waves for nonlinear Schrödinger equations. In this paper, we follow the ideas of [39, 40, 41]. Although the ideas were used before for other problems, the adaptation of the procedure to our problem is not trivial because of the appearance of degenerate Kirchhoff function and the nonlocal nature of the fractional p–Laplacian. For this, we need more delicate estimates and computations.

To show our main results, we first give some notations. For λ > 0, let Wλ be the closure of C0(N)with respect to the norm

uλ=(λ[u]s,pp+up,Vp)1/p,

where

[u]s,p=(2N|u(x)u(y)|p|xy|N+psdxdy)1/p

and

up,V=(NV(x)|u|pdx)1/p.

Then (Wλ , || · ||λ) is a uniformly convex Banach space, see [26] for the details. Moreover, under the condition (V), for each λ > 0 the embedding Wλ ↪ Ws,p(ℝN) is continuous. Indeed, for each uWλ, we have

N|u|pdx{N:V(x)h}|u|pdx+{N:V(x)<h}|u|pdx1h{N:V(x)h}V(x)|u|pdx+|{N:V(x)<h}|pspps({N:V(x)<h}|u|psdx)pps(1h+|{N:V(x)<h}|psppsS1λ1)uλp,

where S > 0 is defined as follows

S=infuDs,pRN0us,ppupsp,

where

|u|ps=(N|u|psdx)1/ps.

Thus,

[u]s,pp+|u|pp(1λ+1h+|{N:V(x)<h}|psppsS1λ1)uλp.

From this it follows that the embedding Wλ ↪ Ws,p(ℝN) is continuous. Next we give the definition of solutions for problem (1.1).

Definition 1.1

We say that uWλ is a (weak) solution of equation (1.1), if

M(uλp)(λ2N|u(x)u(y)|p2(u(x)u(y))|xy|N+ps(φ(x)φ(y))dxdy+NV(x)|u|p2uφdx)=N|u|ps2uφdx+Nf(x,u)φdx,

for any φ ∈ Wλ.

Now we are in a position to state the first result of our paper as follows:

Theorem 1.1

Let (M), (V) and (f1)(f3) hold. Then for any λ > 0, there exists λ* > 0 such that problem (1.1) has a nontrivial solution uλ for any λ ∈ (0, λ*) which satisfies

λ2N|uλ(x)uλ(y)|p|xy|N+psdxdy+NV(x)|uλ|pdx(θpq1m0q1m1θp)1θσ1θλpspsθp,(1.3)

where σ=1q1(1m1m0)+1θp1ps.Assume additionally that f (x, t) is odd with respect to t, for any m ∈ ℕ, there is λm > 0 such that problem (1.1) admits at least m pairs of solutions uλ,i(i = 1, 2, · · · , m) which satisfy (1.3) whenever 0 < λλm.

The proof of Theorem 1.1 is mainly based on the application of the concentration compactness lemma in fractional Sobolev spaces developed by Xiang et al. in [34].We show that the energy functional Iλ associated to problem (1.1) satisfies (PS)c condition for c > 0 small and λ > 0 small. To get the multiplicity of solutions for problem (1.1), we find a special finite dimensional subspaces by which we construct sufficiently small minimax levels. It is worth to point out that the authors in [42] just concerned with the case that M(t) = a+btθ1with a, b > 0, which just focused on the non-degenerate Kirchhoff problems, that is M(0) > 0.

Moreover, for the nonlinear term f , our assumption (f1) is more general than (h1) and (h2) in [42].

Finally, we consider the critical case θ=ps/p.To this aim, we assume the subcritical term f satisfies following assumptions.

(f4) f : ℝN ×ℝ ℝ is a Carathéodory function and there exists q(p,ps)such that for any ε > 0 there exists Cε > 0 and

|f(x,t)|pε|t|p1+qCε|t|q1fora.e.xNandallt;

(f5) There exists q1(p,ps)such that F(x,t)a0|t|q1for a.e. x ∈ ℝN and all t ∈ ℝ.

Theorem 1.2

Assume that M satisfies (M) with θ=ps/pand 2 ≤ p < N/s, and f (x, t) is odd with respect to t and satisfies (f4)− (f5). Then problem (1.1) has infinitely many pairs distinct solutions in Wλ for all λ > 2pSps/p/m0.

For the critical case θ=ps/p,the method used in Theorem 1.1 seems to be invalid. For this, we will use Krasnoselskii’s genus theory to prove Theorem 1.2, see also [43] about the application of the same method to the multiplicity of solutions for a class of fractional Choquard-Kirchhoff equations. Furthermore, as usual for elliptic problems involving critical nonlinearities, we must pay attention to the lack of compactness. To overcome this difficulty, we fix parameter λ larger than a suitable threshold. We would like to point out that the authors in [34] just obtained the existence of two weak solutions for a variant of problem (1.1) by using Ekeland’s variational principle and the mountain pass theorem. To our best knowledge, this is the first time to consider the existence of infinitely many solutions for the critical case θ=ps/pin the study of general Kirchhoff problems.

The rest of our paper is organized as follows. In Section 2, we give the proof of Theorems 1.1. In Section 3, we consider the critical case θ=ps/pand obtain the proof of the Theorem 1.2.

2 Proof of Theorem 1.1

In this section, we prove the main result of this paper. In the following, we shortly denote the norm of Lv(ℝN) by |·|v.

Obviously, the energy functional Iλ : Wλ ℝ associated with problem (1.1)

Iλu=1pMuλp1psRNupsdxRNFx,udx

is well defined, where M(t)=0tM(τ)dτ.It is easy to verify that as argued in [26], IλC1(Wλ , ℝ) and its critical points are solutions of (1.1).

Under our assumptions, we can show that functional has mountain pass geometry.

Lemma 2.1

Assume that (M), (V), and (f1) are satisfied. Then for each λ ∈ (0, 1) there exist αλ > 0 and ρλ > 0 such that Iλ(u) > 0 for uBρλ \ {0}, and Iλ(u) ≥ αλ for all uWλ with ||u||λ = ρλ. Here Bρλ = {uWλ : ||u||λ < ρλ}.

Proof. By (f1), for any ε > 0 there exists Cε > 0 such that

|f(x,t)|εθp|t|θp1+Cεq|t|q1fora.e.xNandallt.

Furthermore, we have

|F(x,t)|ε|t|θp+Cε|t|qfora.e.xNandallt.

For any uWλ, by (M), Hölder’s inequality and the fractional Sobolev inequality, one has

Iλ(u)m0θpuλθp1psN|u|psdxεN|u|θpdxCεN|u|qdx.

Note that by the fractional Sobolev embedding theorem (see [1]), there exists C > 0 such that

|u|θpCuλand|u|qCuλ.

Then choosing ε ∈ (0, m0/(2θpCθp)), we have

Iλ(u)m0θpuλθp1psSps/p[u]s,ppsεCθpuλθpCqCεuλq(m02θp1psSps/puλpsθpCqCεuλqθp)uλθp.

Let us define

g(t):=m02θp1psSps/ptpsθpCqCεtqθpforallt0.

Clearly, limt0+g(t) = m0/(2θp) > 0, since ps>θpand q > θp. Taking ρλ := ||u||λ small enough such that

1psSps/pρλpsθp+CqCερλqθp<m02θp,

then we have

Iλ(u)g(ρλ)ρλθp=:αλ.

Thus we complete the proof.

Lemma 2.2

Under the assumptions of Lemma 2.1, for any finite dimensional subspace E ⊂ Wλ,

Iλ(u)asuE,uλ.

Proof. By (M) and F(x, t) ≥ 0 for a.e. x ∈ ℝN and all t ∈ ℝ, we have

Iλ(u)m1θpuλθp1psN|u|psdx

for all uE. Note that all norms in a finite dimensional space are equivalent. Hence there exists CE > 0 such that |u|psCEuλfor all uE. Then,

Iλ(u)m1θpuλθpa0CEpsuλps.

It follows from ps>θpthat Iλ(u) −∞ as uE, ||u||λ ∞.

Definition 2.1

A sequence {un}n ⊂ Wλ is called a (PS)c sequence, if Iλ(un) → c and Iλ'(un)0.We say Iλ satisfies (PS)c condition if any (PS)c sequence admits a converging subsequence.

Lemma 2.3

Assume that (M) and (f2) are fulfilled. If {un}n is a (PS)c sequence, then {un}n is bounded in Wλ and c ≥ 0.

Proof. Since {un}n is a (PS)c sequence, there exists n0 > 0 such that

Iλ(un)1q1Iλ'(un),unc+o(1)+o(1)unλforallnn0,(2.1)

Then, by (M) and (f2), it follows that

Iλ(un)1q1Iλ'(un),un(m0θpm1q1)unλθp.

Hence, it follows from (2.1) and (m0/(θp) − m1/q1) > 0 that

(m0θpm1q1)unλθpc+o(1)+o(1)unλ.(2.2)

This, together with (m0/(θp) − m1/q1) > 0, yields that {un}n is bounded in Wλ. Then taking the limit in (2.2), we deduce that c ≥ 0. This completes the proof of Lemma 2.3.

Lemma 2.4

Assume that (V), (M) and (f1)− (f2) hold. For any λ ∈ (0, 1), Iλ satisfies the (PS)c condition for all

c(0,σ(m0λθSθ)pspsθp),whereσ=[1q1(1m1m0)+1θp1ps].

Proof. Let {un}n be a (PS)c sequence. Then by Lemma 2.3, {un}n is bounded W, up to a subsequence, there exists a nonnegative function uWλ such that un ⇀ u in W, un → u in Llocσfor σ[1,ps),and un → u a.e. in ℝN. By Theorem 2.2 of [34], up to a subsequence, there exists a (at most) countable set J, a non-atomic measure ζ˜, points {xj}j∈JN and {ζj}j∈J , {ηj}j∈J+ such that as n →

N|un(x)un(y)|p|xy|N+psdyζ=N|u(x)u(y)|p|xy|N+psdy+jJζjδxj+ζ˜(2.3)

and

|un|psη=|u|ps+jJηjδxj(2.4)

in the measure sense, where δxjis the Dirac measure concentrated xj. Moreover,

ηjSps/pζjps/p,jJ,(2.5)

where S > 0 is the best constant of the embedding Ds,p(N)Lps(N).

Next we prove that J = Ø. Otherwise, suppose that J =Ø, then for fixed jJ and ε > 0, choose φε,jC0(N)such that

φε,j=1for|xxj|ε;φε,j=0for|xxj|2ε,

and |▽φε,j| ≤ 2/ε. Evidently, φε,junDs.p(ℝN). Hence it follows from Iλ'(un),φε,jun0that

M(unλp)[λun,unφεs,p+NV(x)|un|pφεdx]=|un|psφε,jdx+Nf(x,un)φε,jundx+o(1),(2.6)

where

un,unφε,js,p=λ2N|un(x)un(y)|p2(un(x)un(y))(φε,j(x)un(x)φε,j(y)un(y))|xy|N+psdxdy.

By using Hölder’s inequality and Lemma 2.3 of [34], we have

limε0limsupnMunεpR2Nunxunyp2unxunyφε,jxφε,jyunxxyN+psdxdyClimε0limsupnR2Nφε,jxφε,jyunxpxyN+psdxdy1/p=0.(2.7)

By (2.3), (2.4) and (M), we have

limε0limsupnMunλpλR2NunxunypxyN+psφε,jydydx+RNVxunpφε,jdxlimεlimnm0λR2NunxunypxyN+psφε,jydxdyθ=m0λζjθ,(2.8)

limε0limnN|un|psφε,jdx=limε0N|u|psφε,jdx+ηj=ηj,(2.9)

and

limε0limnNf(x,un)φε,jundx=limε0Nf(x,u)φε,judx=0.(2.10)

Here we applied the fact that WλLlocv(N)is compact for all v[1,ps).Then we can deduce from (2.6), (2.7), (2.8), (2.9), (2.10) that

ηjm0(λζj)θ.

From the above inequality, together with (2.5), it follows that

ηjm0λθSθηjθp/ps.

Hence,

ηj(m0λθSθ)pspsθp.(2.11)

On the other hand, by (2.3) and (2.4), we obtain

c=limn(Iλ(un)1q1Iλ'(un),un)(m0θpm1q1)(λζj)θ+(1q11ps)ηj[1q1(1m1m0)+1θp1ps](m0λθSθ)pspsθp,

which is a contradiction. Hence the desired conclusion holds.

Letting R > 0, we define

ζ=limRlimsupn{xN:|x|>R}N|un(x)un(y)|p|xy|N+psdydx,

and

η=limRlimsupn{xN:|x|>R}|un|psdx.

In view of Theorem 2.4 of [34], ζ and η are well defined and satisfy

limsupnNN|un(x)un(y)|p|xy|N+psdydx=Ndζ+ζ,(2.12)

and

limsupnNunpsdx=Ndη+η.(2.13)

Assume that χRC(ℝ) satisfies the properties: χR ∈ [0, 1] and XR(x) = 0 for |x| < R, XR(x) = 1 for |x| > 2R, and |▽χR| ≤ 2/R. By Theorem 2.4 of [34], we have

ζ=limRlimsupnN|un(x)un(y)|pχR(x)p|xy|N+psdydx(2.14)

and

η=limRlimsupnN|un(x)χR(x)|psdx.(2.15)

Moreover, we have

Sηp/psζ.(2.16)

Because ||un||p and unpspsare bounded, up to a subsequence, we can assume that ||un||p and unpspsare both convergent. Hence by (2.12) and (2.13), we can obtain

limnunp=Ndζ+ζ(2.17)

and

limnunpsps=Ndη+η.(2.18)

It follows from Iλ'(un),χRun0as n → ∞ that

M(unλp)[λ2N|un(x)un(y)|pχRdx|xy|N+psdxdy+NV(x)|un|pχRdx]+λ2N|un(x)un(y)|p2(un(x)un(y))un(y)(χR(x)χR(y))|xy|N+psdxdy=N|un|psχRdx+Nf(x,un)unχRdx+o(1).(2.19)

By employing Hölder’s inequality and (2.15) in [34], we get

limRlimsupn2N|un(x)un(y)|p2(un(x)un(y))un(y)(χR(x)χR(y))|xy|N+psdxdy=0.(2.20)

Hence we deduce from (2.14), (2.17), (2.19) and (2.20) that

limRlimsupnMunλpλR2NunxunypχRxxyN+psdxdy+RNVxunpχRdxm0λθRNdζ+ζθ1limRlimsupnxRN:x>RRNunxunypxyN+psdydxm0λθζθ,(2.21)

thanks to the assumption θ > 1. It is easy to see that

limRlimsupnNf(x,un)unχRdx=limRNf(x,u)uχRdx=0,(2.22)

Thanks to the fact that the embedding WλLlocq(N)is compact. Therefore, we conclude from (2.19), (2.20), (2.21), (2.22) and (2.15) that

m0λθζθη,

which together with (2.16) yields

m0λθSθηθp/psη.

This implies that η = 0 or

η(m0λθSθ)pspsθp.(2.23)

Assume that (2.23) holds. Then

c=limnIλun1q1Iλun,unlimnm0θpm1q1RNunxunypxyN+psdxdyθ+1q1+1psRNunpsdx1q11m1m0+1θp1psm0λθSθpspsθp,

which is absurd. Hence, we have ν = 0. In view of J = Ø and (2.18), we have

limnN|un|psdx=N|u|psdx.(2.24)

Now we show that un → u in Wλ. To this aim, we first assume that d := infn≥1 ||un||λ > 0.

Since Iλ'(un)Iλ'(u),unu0,we have

M(unλp)un,unuM(uλp)u,unu=N(|un|ps2un|u|ps2u)(unu)dx+o(1),

where

un,unu:=λun,unus,p+NV(x)|un|p2un(unu)dx.

Here we used the following fact:

limnN(f(x,un)f(x,u))(unu)dx=0.(2.25)

Now we show that (2.25) is true. By Theorem 2.1 of [26], we know that the embeddingWλ Lν(ℝN) is compact for any v[p,ps).Thus, up to a subsequence, we have un u in Lν(ℝN) for any v[p,ps).According to (f1) and (f2), for any ε > 0 we have

|f(x,t)|ε|t|θp1+Cε|t|q1

for all (x, t) ∈ ℝN × ℝ. Then

|N(f(x,un)f(x,u))(unu)dx|εN(|un|θp1+|u|θp1)(unu)dx+CεN(|un|q1+|u|q1)(unu)dxCε+Cε(unqq1+uqq1)unuq,

which implies that

limnNfx,unfx,uunudx=0.

Thus, we obtain

Munλpun,unuu,unu+Munλpun,unuMunλpu,unu=Nunps2unups2uunudx+o1.

By the boundedness of {un}n and un u in Wλ, we can deduce that

limnMunλpu,unu=0

and

limnMuλpu,unu=0.

Hence, we conclude from (2.24) that

limnMunλpun,unuu,unu=0.

This, together with d := infn≥1 ||un|| > 0, implies that

limnun,unuu,unu=0.(2.26)

Let us now recall the well-known inequalities:

abp2pap2abp2babforp21p1ap2abp2babp/2ap+bp2p/2for1<p<2,(2.27)

for all a, b ∈ ℝN.

Similar to the proof of Lemma 6 in [26], it is easy to deduce from (2.27) that un → u strongly in Wλ as n → ∞.

In the end, we consider the case infnunλ=0.Then either 0 is an accumulation point of the sequence {un}n and so there exists a subsequence of {un}n strongly converging to u = 0, or 0 is an isolated point of the sequence {un}n and so there exists a subsequence, still denoted by {un}n, such that infn ||un|| > 0. In the first case we are done, while in the latter case we can proceed as above.

Since the functional Iλ satisfies the (PS)c condition for small c > 0, we will find a special finite dimensional subspaces by which we construct sufficiently small minimax levels.

By (V), we know that Vx0=minxNV(x) = 0. Without loss of generality, we assume from now on that x0 = 0. By means of (M) and (f3), we have

Iλum1θpuλθp1psNupsdxa0Nuq2dxm1θpuλθpa0Nuq2dx

for all uWλ. Define the functional Фλ : Wλ ℝ by

Jλ(u)=m1θpuλθpa0N|u|q2dx.

Then Iλ(u) ≤ Jλ(u) for all uWλ. Hence it suffices to construct small minimax levels for Jλ.

For any δ > 0, one can choose ϕδC0(N)with ||фδ||q2 = 1 and supp фδBrδ (0) such that t[ϕδ]s,pp<δ.

Let

eλ=ϕδλθpsNpsθpx.

Then supp eλBθpsλN(psθp)rδ(0).Thus, for t ≥ 0, we have

Jλ(teλ)=tθpθpeλλθpa0tq2N|eλ|q2dx=λθpspsθptθp1θp[(2N|ϕδ(x)ϕδ(y)|p|xy|N+psdxdy+λ(θ1)θpspsθpNV=(λθpsN(psθp)x)|ϕδ|pdx)θtq2a0N|ϕδ|q2dx].

It follows from λ ∈ (0, 1) and θ > 1 that

Jλteλλθpspsθptθp1θp2NϕδxϕδypxyN+psdxdy+NV(λθpsNpsθpx)ϕδpdxθa0tq2Nϕδq2dx=λθpspsθpΦλtϕδ,

where ФλC1(Wλ , ℝ) defined by

Φλu=1θp2NuxuypxyN+psdxdy+NV(λθpsNpsθpx)updxθa0Nuq2dx

for all uWλ. Clearly,

maxt0Φλtϕδ=qθpθpqqa0θpqθpR2NϕδxϕδypxyN+psdxdy+RNVλθpsNpsθpxϕδpdxqqθp.

Observe that V(0) = 0 and VC(ℝN), then there exists Λδ > 0 such that

0V(λθpsN(psθp)x)δϕδpp

for all |x| ≤ rδ and 0 < λΛδ. It follows from [ϕδ]s,pp<δthat

maxt0Φλ(tϕδ)q2θpθpq2(q2a0)θpq2θp(2δ)q2q2θp.

Furthermore, we have

maxt0Iλ(tϕδ)q2θpθpq2(q2a0)θpq2θp(2δ)q2q2θpλθpspsθp,(2.28)

for all λ ∈ (0, Λδ]. In conclusion, we have the following lemma.

Lemma 2.5

Under the assumptions of Lemma 2.1, there exists Λ > 0 such that for all λ ∈ (0, Λ) there exists e˜λWλwith e˜λλ>ρλ,Iλ(e˜λ)<0and

maxt[0,1]Iλ(te˜λ)<σλθpspsθp,

where σ=1q1(1m1m2)+1θp1ps.

Proof. Let δ > 0 small enough such that

q2θpθpq2(q2a0)θpq2θp(2δ)q2q2θp<σ.

Taking Λ = Λδ and choosing tλ~>0such that t˜λeλλ>ρλand Iλ(teλ)< 0 for all tt˜λThe result follows by letting e˜λ=t˜λeλ.

Let m ∈ ℕ, we choose m functions ϕδiC0(N)such that supp ϕδisupp ϕδj=for all 1 ≤ ijm, and ||ϕδ||q = 1 and ϕδs,pp<δ.Let rδm> 0 be such that supp ϕδi⊂ Brδm(0) for i = 1, 2, · · · , m. Set

eλi=ϕδiλθpSNpSθpxforall i=1,2,...,m,

and

Eλ,δm=spaneλ1,eλ2,...,eλm.

Then for each u=i=1mcieλiEλ,δm,we have

2NuxuypxyN+psdxdy=i=1mcip2NeλixeλiypxyN+psdxdy,

NVxupdx=i=1mcipNVxeλipdx,

1psNupsdx=1psi=1mcipsNeλipsdx

and

NFx,udx=i=1mNFx,cieλidx.

Hence,

Iλu=i=1mIλcieλi.

and as above

IλcieλiλθpSNpSθpΦλcieλi.

Set

βδ:=maxϕδipp:i=1,2,...,m,

and choose Λm,δ > 0 such that

VλθpSNpSθpxδβδ

for all 1xrδmand λΛm,δ. As in the proof of Lemma 2.4, we can get

maxuEλ,δmIλumq2θpθpq2q2a0θpq2θp2δq2q2θpλθpspsθp,

for all λ ∈ (0, Λm,δ]. Then we have the following lemma.

Lemma 2.6

Under the assumptions of Lemma 2.1, for any m ∈ ℕ there exists Λm > 0 such that for all λ ∈ (0, Λm) there exists m-dimensional subspace Eλmsuch that

maxuEλmIλu<σλθpspsθp.

Proof. Choose δ > 0 so small that

q2θpθpq2q2a0θpq2θp2δq2q2θp<σ

and take Eλm=Eλ,δm.The result follows from (2.28) and the definition of Eλ,δm.

Proof of Theorem 1.1

According to Lemma 2.5, we choose Λσ > 0 and define

cλ=infyΓmaxt0,1Iλte˜λ,

where

Γ=yC0,1,Wλ:y0=0andy1=e˜λ.

By Lemma 2.1, we have αλcλ < σλθps/psθp.In view of Lemma 2.4, we know that Iλ satisfies the (PS) condition, and there exists uλ ∈ Wλ such that Iλ(uλ)=0and Iλ(uλ) = cλ. Thus, uλ is a solution of (1.1).

It follows from Iλ(uλ)=0and Iλ(uλ) = cλ that

σλθps*ps*θpIλ(uλ)=Iλ(uλ)1q1I(uλ),uλ(m0θpm1q1)uλλθ+(1q11ps*)N|uλ|ps*dx.

Hence (1.3) holds.

Denote the set of all symmetric (in the sense that −Z = Z) and closed subsets of E by Σ, for each Z ∈ Σ. Let y(Z) be the Krasnoselski genus and

i(Z)=minhΓmy(h(Z)Bρλ),

where Γm is the set of all odd homeomorphisms h ∈ C(Wλ ,Wλ) and ρλ is the number from Lemma 2.1. Then i is a version of Benci’s pseudo-index (see [44]). Let

cλj:=infi(Z)jsupuZIλ(u),1jm.

Since Iλ(u) ≥ αλ for all u∂Bρλ and i(Eλ,m) = dim Eλ,m = m, we have

αλcλ1cλmsupuEλ,mIλ(u)<[1q1(1m1m0)+1θp1ps]λθpspsθp.

It follows from Lemma 2.4 that Iλ satisfies the (PS)c condition at all levels cλj(j = 1, 2, · · · , m). According to standard critical point theory (see [45]), all cλjare critical values, and then Iλ has at least m pairs of nontrivial critical points.

3 Proof of Theorem 1.2

In this section, we consider the existence of infinitely many solutions of problem (1.1), where the Kirchhoff function M satisfies (M) with the critical case θ=ps/p.Let us first recall some basic results about Krasnoselskii’s genus, which can be found in [45]. Let Gbe a real Banach space. Set

Γ={AG\{0}:AiscompactandA=A.}

Definition 3.1

Let A ∈ Γ. The Krasnoselskii genus (A) of A is defined as being the least positive integer k such that there is an odd mapping фC(A, ℝk) such that ф (x) ≠0 for all xA. If such a k does not exist we set (A) = ∞. Moreover, by definition, (Ø) = 0.

Lemma 3.1

(see [45]) Let G= ℝN and ∂Ω be the boundary of an open, symmetric, and bounded subset Ω ⊂N with 0 ∈ Ω. Then (∂Ω) = N.

Denote by 𝕊N−1 the surface of the unit sphere in ℝN. Then we can deduce from Lemma 3.1 that (𝕊N−1) = N.

We shall use the following theorem to obtain the existence of infinitely many solutions for (1.1).

Theorem 3.1

(see [46]) Let TC1(G, ℝ) be an even functional satisfying the (PS) condition. Furthermore,

  • (1)

    T is bounded from below and even;

  • (2)

    there is a compact set EΓ such that (E) = k and supuE T(u) < T(0).

Then T has at least k pairs of distinct critical points and their corresponding critical values are less than T(0).

Lemma 3.2

Assume that s ∈ (0, 1), 2 ≤ p < N/s, θ=ps/p,p < q <psand f satisfies (f4). Then functional Iλ satisfies the (PS)c conditions in Wλ for all λ > 2pSps/p/m0.

Proof. Let {un}nWλ be the (PS)c sequence of functional Iλ, i.e.

Iλ(un)c,Iλ(un)0

as n → ∞.

By (f4), we have

|F(x,t)||t|p+C1|t|qfora.e.xNandallt.

It follows from (M), (f4), Hölder’s inequality and the fractional Sobolev inequality that

Iλ(u)m0θpuλθp1psN|u|psdxN|u|pdxC1N|u|qdxm0θpuλθp1psSpspλ1uλpsCuλpuλq,(3.1)

for all uWλ. When θ=ps/p,since λ>2pSps/pm0>Sps/ppsθpm0and p < q < ps,it is easy to see that Iλ is coercive and bounded from below on Wλ. Hence, {un}n is bounded in Wλ. Then there exist a subsequence of {un}n (still denoted by {un}n) and uWλ such that

unuinWλandinLps(N),

unua.e.inN,

|un|ps2un|u|ps2uinLpsps1(N),(3.2)

as n → ∞. Similar to the discussion as in Section 2, we have

limnN(f(x,un)f(x,u))(unu)dx=0.(3.3)

Let wn = unu. Then by using similar arguments as in Lemma 3.2 of [4], we get

[un]s,pp=[wn]s,pp+[u]s,pp+o(1).(3.4)

By the celebrated Brézis–Lieb lemma, one has

NV(x)|wn|pdx=NV(x)|un|pdxNV(x)|u|pdx+o(1)(3.5)

and

N|wn|psdx=N|un|psdxN|u|psdx+o(1).(3.6)

Let us now introduce, for simplicity, for all vWλ the linear functional ℒ(v) on Wλ defined by

L(v),w=λ2N|v(x)v(y)|p2(v(x)v(y))(w(x)w(y))|xy|N+psdxdy+NV(x)|v|p2vwdx

for all wWλ. The Hölder inequality gives that

Lv,wλvs,pp1ws,p+RNVxvpdxp1/pRNVxwpdx1/pλvs,pp1+RNVxvpdxp1/pwλ

Thus, for each vWλ, the linear functional ℒ(v) is continuous on Wλ. Hence, the weak convergence of {un}n in Wλ gives that

limnL(u),unu=0.(3.7)

Without loss of generality, we assume that limn||wn||λ = η. Since {un}n is a (PS)c sequence, by the boundedness of {un}n, (3.3), (3.6) and (3.7), we have

o(1)=Iλ(un)Iλ(u),unu=M(unλp)L(un),unuM(uλp)L(u),unuN[f(x,un)f(x,u)](unu)dxN[|un|ps2un|u|ps2u](unu)dx

=Mun λ pLun,unuLu,unuRNunupsdx+o1.(3.8)

Here we use the following fact:

limnN|un|ps2unudx=N|u|psdx,(3.9)

thanks to (3.2).

It follows from (3.4), (3.5) and (3.8) that

M(unuλp+uλp)[L(un),unuL(u),unu]N|unu|psdx+o(1).(3.10)

From the definition of S, we get

N|wn|psdxSpsp[wn]s,pps.

Putting this in (3.10) and using (M) and (2.27) with the case p ≥ 2, we arrive at the inequality

m0(unuλp+uλp)(θ1)p12punuλpSps/pλ1unuλps+o(1).

Letting n → ∞, we have

12pm0(ηp+uλp)(θ1)pηpSps/pλ1ηps.

This implies that

m0ηθpλ12pSps/pηps.(3.11)

Since θ=ps/pand 2pSps/p/m0<λ,it follows from (3.11) that η = 0. Thus, un → u in Wλ.

Remark 3.1

It seems that the method used in the proof of Lemma 3.2 could not be applied to the case θ > ps/p.

Proof of Theorem 1.2

Denote by {e1, e2, · · · } a basis of Wλ, and for each k ∈ ℕ consider εk = span{e1, e2, · · · , ek}, the subspace of Wλ generated by e1, e2, · · · , ek. By assumption p<q<ps,we know that εk can be continuously embedded into Lq(ℝN).Note that all norms are equivalent on a finite dimensional Banach space. Thus there exists a positive C(k) depending on k such that

uλqC(k)N|u|qdx,

for all u ∈ εk. Then by (M1) and (f5), we deduce

Iλ(u)m1puλθp1psN|u(x)|psdxa0C(k)uλq(m1puλθpqa0C(k))uλq,

for all u ∈ εk. Let R be a positive constant such that

m1pRθpq<a0C(k).

Hence, for all 0 < r < R, we get

Iλ(u)rq(m1prθpqa0C(k))Rq(m1pRθpqa0C(k))<0=J(0),

for all u ∈ K := {u ∈ εk : ||u||λ = r}. It follows that

supuKIλ(u)<0=Iλ(0).

Clearly, εk and ℝk are isomorphic and 𝒦 and 𝕊k−1 are homeomorphic. Thus, we conclude that y(𝒦) = k by Lemma 3.1. Since f (x, u) is odd with respect to u ∈ ℝ, the functional Iλ is even. Moreover, by (3.1), we know Iλ is bounded from below and satisfies the (PS)c condition by Lemma 3.2. It follows from Theorem 3.1 that Iλ has at least k pairs of distinct critical points. The arbitrariness of k yields that Iλ has infinitely many pairs distinct critical points in Wλ, that is, problem (1.1) has infinitely many pairs distinct solutions.

Acknowledgement

M. Xiang was supported by the National Natural Science Foundation of China (No. 11601515) and Tianjin Key Lab for Advanced Signal Processing (No. 2016ASP-TJ02). B. Zhang was supported by the National Natural Science Foundation of China (No. 11871199). V.D. Rădulescu acknowledges the support through the Project MTM2017-85449-P of the DGISPI (Spain).

References

  • [1]

    E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136, (2012), no. 5, 521–573. Google Scholar

  • [2]

    T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on ℝN Commun. Partial Differ. Equ. 20, (1995), no. 9-10, 1725–1741. CrossrefGoogle Scholar

  • [3]

    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

  • [4]

    R. Servadei and E. Valdinoci, The Brézis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367, (2015), no. 1, 67–102. Google Scholar

  • [5]

    R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, Revista Matemática Complutense 28, (2015), no. 3, 655–676. CrossrefGoogle Scholar

  • [6]

    X. Ros-Oston and J. Serra, Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Comm. Partial Differential Equations 40, (2015), no. 1, 115–133. CrossrefGoogle Scholar

  • [7]

    G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in ℝN J. Differential Equations 255, (2013), no. 8, 2340–2362. CrossrefWeb of ScienceGoogle Scholar

  • [8]

    D. Applebaum, Lévy processes–from probability to finance quantum groups, Notices Amer. Math. Soc. 51, (2004), no. 11, 1336–1347. Google Scholar

  • [9]

    L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia 7 (2012) 37– 52. CrossrefGoogle Scholar

  • [10]

    G. Molica Bisci and D. Repovš, Higher nonlocal problems with bounded potential, J. Math. Anal. Appl. 420, (2014), no. 1, 591–601. Google Scholar

  • [11]

    G. Molica Bisci and V. Rădulescu, Ground state solutions of scalar field fractional for Schrödinger equations, Calc. Var. Partial Differential Equations 54, (2015), no. 3, 2985–3008. CrossrefGoogle Scholar

  • [12]

    G. Molica Bisci and V. Rădulescu, Multiplicity results for elliptic fractional equations with subcritical term, Nonlinear Differential Equations Appl. NoDEA, 22, (2015), no. 4, 721–739. CrossrefGoogle Scholar

  • [13]

    G. Molica Bisci, V. Rădulescu, R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016. Google Scholar

  • [14]

    N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268, (2000), no. 4-6, 298-305. CrossrefGoogle Scholar

  • [15]

    N. Laskin, Fractional Schrödinger equation, Phys. Rev. E 66, (2002), 056108. Google Scholar

  • [16]

    G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401, (2013), no. 2, 706–713. CrossrefWeb of ScienceGoogle Scholar

  • [17]

    X. M. He and W. M. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl. 193, (2014), no. 2, 473–500. CrossrefGoogle Scholar

  • [18]

    A. Ourraoui, On a p–Kirchhoff problem involving a critical nonlinearity, C. R. Math. Acad. Sci. Paris Ser. I 352, (2014), no. 4, 295–298. CrossrefGoogle Scholar

  • [19]

    Y. He, G. B. Li, S. J. Peng, Concentrating bound states for Kirchhoff type problems in R3 involving critical Sobolev exponents, Adv. Nonlinear Stud. 14, (2014), no. 2, 483–510. Google Scholar

  • [20]

    J. Liu, J. F. Liao, C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in ℝN J. Math. Anal. Appl. 429, (2015), no. 2, 1153–1172. CrossrefWeb of ScienceGoogle Scholar

  • [21]

    S. H. Liang and S. Y. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in ℝN Nonlinear Anal. 81, (2013), 31–41. CrossrefWeb of ScienceGoogle Scholar

  • [22]

    A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonliear Anal. 94, (2014), 156– 170 CrossrefGoogle Scholar

  • [23]

    P. Pucci, M. Q. Xiang, B. L. Zhang, Existence and multiplicity of entire solutions for fractional p–Kirchhoff equations, Adv. Nonlinear Anal. 5, (2016), no. 1, 27–55. Google Scholar

  • [24]

    S. H. Liang, D. Repovš, B. L. Zhang, On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity, Comput. Math. Appl. 75, (2018), no. 5, 1778–1794. Google Scholar

  • [25]

    P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in ℝN involving nonlocal operators, Rev. Mat. Iberoam 32, (2016), no. 1, 1–22. CrossrefGoogle Scholar

  • [26]

    P. Pucci, M. Q. Xiang, B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p–Laplacian in ℝN Calc. Var. Partial Differential Equations 54, (2015), no. 3, 2785–2806. CrossrefGoogle Scholar

  • [27]

    M. Q. Xiang, B. L. Zhang, M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non–local fractional p–Laplacian, J. Math. Anal. Appl. 424, (2015), no. 2, 1021–1041. CrossrefWeb of ScienceGoogle Scholar

  • [28]

    M. Q. Xiang, B. L. Zhang, M. Ferrara, Multiplicity results for the nonhomogeneous fractional p–Kirchhoff equations with concave-convex nonlinearities, Proc. Roy. Soc. A 471, (2015), no. 2177, 14 pp. Google Scholar

  • [29]

    X. Mingqi, G. Molica Bisci, G. H. Tian, B. L. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p–Laplacian, Nonlinearity 29, (2016), no. 2, 357–374. CrossrefWeb of ScienceGoogle Scholar

  • [30]

    M. Q. Xiang, B. L. Zhang, V. Rădulescu, Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p–Laplacian, Nonlinearity 29, (2016), no. 10, 3186–3205. CrossrefWeb of ScienceGoogle Scholar

  • [31]

    Z. Binlin, A. Fiscella, S. Liang, Infinitely many solutions for critical degenerate Kirchhoff type equations involving the fractional p–Laplacian, Appl. Math. Optim. doi: 10.1007/s00245-017-9458-5. Web of ScienceGoogle Scholar

  • [32]

    G. Autuori, A. Fiscella, P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal. 125, (2015), 699–714. Web of ScienceCrossrefGoogle Scholar

  • [33]

    M. Q Xiang, B. L. Zhang, H. Qiu, Existence of solutions for a critical fractional Kirchhoff type problem in ℝN Sci. China Math. 60, (2017), no. 9, 1647–1660. CrossrefGoogle Scholar

  • [34]

    M. Q. Xiang, B. L. Zhang, X. Zhang, A nonhomogeneous fractional p–Kirchhoff type problem involving critical exponent in ℝN Adv. Nonlinear Stud. 17, (2017), no. 3, 611–640. Google Scholar

  • [35]

    P. L. Lions, The concentration-compactness principle in the calculus of variations, the limit case, Part I. Rev. Mat. Iberoam. 1, (1985), 145–201. [Erratum in Part II, Rev. Mat. Iberoam. 1, (1985), 45–121.] Google Scholar

  • [36]

    A. Fiscella and P. Pucci, p–fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl. 35, (2017), 350–378. CrossrefGoogle Scholar

  • [37]

    J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrodinger equations, Arch. Ration. Mech. Anal. 165, (2002), no. 11-12, 295–316. CrossrefGoogle Scholar

  • [38]

    D. Cao and E. S. Noussair, Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations, J. Differ. Equations 203, (2004), no. 2, 292–312. CrossrefGoogle Scholar

  • [39]

    Y. H. Ding and F. H. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differentail Equations 30, (2007), no. 2, 231–249. CrossrefGoogle Scholar

  • [40]

    Y. H. Ding and J. C. Wei, Semiclassical states for Schrödinger equations with sign-changing potentials, J. Funct. Anal. 251, (2007), no. 2, 546–572. CrossrefGoogle Scholar

  • [41]

    S. H. Liang and J. H. Zhang, On some p–Laplacian equation with electromagnetic fileds and critical nonlinearity in ℝN J. Math. Phys. 56, (2015), 041504. CrossrefGoogle Scholar

  • [42]

    Y. Q. Song and S. Y. Shi, Solutions of p–Kirchhoff problems with critical nonlinearity in ℝN J. Nonlonear Sci. Appl. 11, (2018), no. 2, 172–188. CrossrefGoogle Scholar

  • [43]

    F. Wang and M. Q. Xiang, Multiplicity of solutions for a class of fractional Choquard-Kirchhoff equations involving critical nonlinearity, Anal. Math. Phys. (2017). https://doi.org/10.1007/s13324-017-0174-8 Web of Science

  • [44]

    V. Benci, On critical point theory of indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc. 274, (1982), no. 2, 533–572. CrossrefGoogle Scholar

  • [45]

    P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986. Google Scholar

  • [46]

    D. C. Clarke, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J. 22, (1972), no. 1, 65–74. Google Scholar

About the article

Received: 2018-11-18

Accepted: 2019-02-15

Published Online: 2019-08-06

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 690–709, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0021.

Export Citation

© 2020 Mingqi Xiang et al., published by De Gruyter. 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.

[1]
Antonio Iannizzotto, Sunra Mosconi, and Marco Squassina
Nonlinear Analysis, 2020, Volume 191, Page 111635
[2]
Li Wang, Kun Cheng, and Binlin Zhang
Applied Mathematics & Optimization, 2019
[3]
Mingqi Xiang, Di Yang, and Binlin Zhang
Asymptotic Analysis, 2019, Page 1
[4]
Mingqi Xiang, Di Yang, and Binlin Zhang
Complex Variables and Elliptic Equations, 2019, Page 1

Comments (0)

Please log in or register to comment.
Log in