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Advances in Nonlinear Analysis

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Existence and concentration behavior of solutions for a class of quasilinear elliptic equations with critical growth

Kaimin Teng
  • Corresponding author
  • Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, P. R. China
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/ Xiaofeng Yang
Published Online: 2017-04-19 | DOI: https://doi.org/10.1515/anona-2016-0218

Abstract

In this paper, we study a class of quasilinear elliptic equations involving the Sobolev critical exponent

-εpΔpu-εpΔp(u2)u+V(x)|u|p-2u=h(u)+|u|2p*-2uin N,

where Δpu=div(|u|p-2u) is the p-Laplace operator, p*=NpN-p (N3, N>p2) is the usual Sobolev critical exponent, the potential V(x) is a continuous function, and the nonlinearity h(u) is a nonnegative function of C1 class. Under some suitable assumptions on V and h, we establish the existence, multiplicity and concentration behavior of solutions by using combing variational methods and the theory of the Ljusternik–Schnirelman category.

Keywords: Quasilinear Schrödinger equation; mountain pass theorem; ground state solution, Ljusternik–Schnirelman category

MSC 2010: 35J20; 35J62

1 Introduction

In this paper, we are concerned with the existence, multiplicity and concentration behavior of solutions of the following quasilinear elliptic equations involving the Sobolev critical exponent:

-εpΔpu-εpΔp(u2)u+V(x)|u|p-2u=h(u)+|u|2p*-2uin N,(1.1)

where Δpu=div(|u|p-2u) is the p-Laplacian, N3, 2p<N, p*=NpN-p is the Sobolev critical exponent, ε>0 is a small parameter, the potential V:N is a continuous function, and the nonlinearity h(u): is a nonnegative function of C1 class. The reduction form of equation (1.1) appears in many branches of mathematical physics and has been studied extensively in recent years. In particular, when p=2, the solution of (1.1) is related to the following quasilinear Schrödinger equation:

iεψt=-ε2ψ+W(x)ψ-κε2(ρ(|ψ|2))ρ(|ψ|2)ψ-l~(|ψ|2)ψ,(1.2)

where ψ:×N, W:N is a given potential, κ is a real constant, and ρ, l are real functions. Equation (1.2) arises in several models of different physical phenomena corresponding to various types of ρ. The case ρ(s)=s is used for the superfluid film equation in plasma physics by Kurihura in [26]. In the case ρ(s)=(1+s)12, it models the self-channeling of a high-power ultra short laser in matter; see [8, 9, 12, 15]. For more physical motivations, we can refer the interested readers to [6, 23, 25] and the references therein. If the quasilinear term εpΔp(u2)u is not appearing and p>2, problem (1.1) arises in a lot of applications when ε=1, such as image processing, non-Newtonian fluids and pseudo-plastic fluids; for more details see [5, 13, 19] and the references therein.

When κ=1 and ρ(s)=s, considering standing wave solutions of the form ψ(x,t)=u(x)e-iEt/ε in (1.2), then u(x) verifies the following equation:

-ε2Δu-ε2Δ(u2)u+V(x)u=l(u)in N,(1.3)

where V(x)=W(x)-E, l(u)=l~(u2)u. It is clear that when ε=1, equation (1.3) reduces to the following equation:

-Δu-Δ(u2)u+V(x)u=l(u)in N.(1.4)

When κ=0 and ρ(s)=s, the standing wave solutions of equation (1.2) satisfies the classical Schrödinger equation of the form

-ε2Δu+V(x)u=l(u)in N.

The existence and concentration behavior of positive solutions of the above equation have been extensively investigated under various hypotheses on the potential V(x) and the nonlinearity l(u); see, for example, [4, 16, 17, 22, 32, 34] and the references therein.

In recent years, many scholars have been interested in the study of the existence and multiplicity of solutions for equation (1.4). For example: the existence of a positive ground state solution has been obtained in [33] by using a constrained minimization argument which gives a solution of (1.4) with an unknown Lagrange multiplier λ in front of the nonlinear term. In [30], the existence of both one-sign and nodal ground states of soliton type solutions were established by the Nehari manifold method. In [29], Liu, Wang and Wang developed the methods of change of variables such that the quasilinear problem reduces to a semilinear one. They used an Orlicz space framework to prove the existence of positive solutions of (1.4) by the mountain pass theorem. Meanwhile, Colin and Jeanjean [14] developed the dual methods to treat the quasilinear problem (1.5), and the usual Sobolev space H1(N) was used to prove the existence of positive solutions. The other recently interesting works can be found in [3, 11, 20, 18, 31] and the references therein.

Regarding critical problems, there are also some important results appearing in the literature. For example, in [24], the authors established the existence, multiplicity and concentration behavior of ground states for quasilinear Schrödinger equation with critical growth

{-ε2Δu-ε2Δ(u2)u+V(x)u=h(u)+|u|22*-2uin N,uH1(N),u(x)>0,(1.5)

by using the variational methods and combining them with the theory of the Ljusternik–Schnirelman category which was used by Alves, Figueiredo and Severo [2] to establish the existence and multiplicity of nontrivial weak solutions for quasilinear elliptic equations of the form

-εpΔpu-εpΔp(u2)u+V(x)|u|p-2u=h(u)in N.

Yang and Ding [40] applied the perturbed methods to consider the following critical quasilinear Schrödinger equation:

-ε2Δu-ε2Δ(u2)u+V(x)u=h(x,u)u+K(x)|u|22-2uin N,

and showed the existence of positive solutions as εε0 and for any m; it has at least m pairs of solutions if εεm, where ε0 and εm are sufficient small positive numbers.

Unlike [24, 2, 40], where the minimum of V(x) is global, Wang and Zou [38] studied the quasilinear Schrödinger equation with critical exponent

-ε2u-ε2[(u2)]u+V(x)u=g(u)+|u|22*-2uin N,

where the potential V(x) satisfies the local minimum condition infΩV<infΩV, Ω is a bounded domain of N, and proved the existence of positive bound states which concentrate around the local minimum point of V.

Motivated by the above-cited papers, the main purpose of this paper is to establish the existence, multiplicity and concentration behavior of the ground states for the quasilinear elliptic equation with critical growth

{-εpΔpu-εpΔp(u2)u+V(x)|u|p-2u=h(u)+|u|2p*-2uin N,uW1,p(N),u(x)>0in N,(Pe)

where V:N is a continuous function satisfying

  • (V)

    0<V0=infxNV(x)<lim|x|infV(x)=V<.

Assume that the nonlinearity h: is of class C1 and satisfies the following conditions:

  • (H0)

    h(s)=0 for s0, h(s)=o(|s|p-2) as s0;

  • (H1)

    lim|s|h(s)|s|q-1=0 for some q(2p-1,2p*-1);

  • (H2)

    there exists 2p<θ<2p* such that 0<θH(s)=θ0sh(τ)dτsh(s) for all s>0;

  • (H3)

    the function sh(s)/s2p-1 is increasing for s>0;

  • (H4)

    there exist C>0, σ(max{2pN/(N-p)-2NN-p,2p},2p*) such that h(s)Csσ-1 for s>0.

As far as we know, little work has been done for the existence and concentration behavior of positive solutions for the quasilinear problem (Pe) where the nonlinearity has a critical growth. Our main result complements the corresponding conclusion of [2] and extends the main result of [24]. Alves, Figueiredo and Severo [2], He, Qian and Zou [24] and Wang and Zou [38] chose the Sobolev space E which is defined by

E={vW1,p(N):NV(x)|f(v)|pdx<}

equipped with the norm

vE=vLp+|v|f:=vLp+infξ>01ξ[1+NV(x)|f(ξv)|pdx].

The Orlicz–Sobolev space E may not be reflexible for p=2, and so the bounded sequence may have no convergent subsequence in E. Unlike the work of [2, 24, 38], we directly choose the usual Sobolev space W1,p(N) to deal with the autonomous problem and the usual Sobolev space X which will be defined in Section 2 to treat the nonautonomous problem. On the other hand, we use the mountain pass theorem under (C)c condition (see [36]); this is different from [2, 24, 38].

For stating our main result, we set

M={xN:V(x)=V0}

and

Mδ={xN:dist(x,M)δ}for δ>0.

In view of (V), the set M is compact. We recall that, if Y is a closed subset of a topological space X, the Ljusternik–Schnirelman category catX(Y) is the least number of closed and contractible sets in X which cover Y. By means of the Ljusternik–Schnirelman theory, we arrive at the following result.

Theorem 1.1.

Suppose that conditions (V) and (H0)(H5) are satisfied. Given δ>0, there exists ε¯=ε¯(δ)>0 such that for any ε(0,ε¯), problem (Pe) has at least catMδ(M) positive weak solutions in Cloc1,α(RN)L(RN). Moreover, each solution decays to zero at infinity, and if uε denotes one of these positive solutions and ηεRN its global maximum, then

limε0V(ηε)=V0.

The paper is organized as follows: In Section 2, we present the abstract framework of the problem as well as some preliminary results. In Section 3, we show the existence of ground state solution for autonomous problem. Section 4 is devoted to the proof of Theorem 1.1.

2 Variational framework and preliminary results

Formally, the energy functional associated to (Pe) is defined by

I(u)=εppN(1+2p-1|u|p)|u|pdx+1pNV(x)|u|pdx-NH(u)dx-12p*N|u+|2p*dx,(2.1)

where H(u)=0uh(s)ds, u+=max{u,0}. Observe that p(u2)u is not always in L1(N), therefore the functional I(u) is not well defined on the whole Sobolev space W1,p(N). In fact, let uC01(N{0}) and u(x)=|x|p-N/2p, xB1{0}; then uW1,p(N), while the function |u|p|u|pL1(N). To overcome this difficulty, we use the change of variable methods developed in [29], making the change of variables u=f(v), where f is a C function and defined by

f(t)=1(1+2p-1|f(t)|p)1pfor t[0,+)

and

f(-t)=-f(t)for t(-,0].

The following properties were proved in [35].

Lemma 2.1.

The following properties involving f and its derivative hold:

  • (1)

    f is a uniquely defined C function and invertible;

  • (2)

    0<f(t)1 for all t ;

  • (3)

    f(t)t1 as t0 ;

  • (4)

    12f(t)tf(t)f(t) for all t0 , and 12f2(t)tf(t)f(t)f2(t) for all t ;

  • (5)

    the function f(t)t is decreasing for t>0 ;

  • (6)

    |f(t)||t| for all t ;

  • (7)

    |f(t)f(t)|1/(2p-1p)<1 for all t ;

  • (8)

    f(t)/|t| is nondecreasing for t>0 and limt+f(t)/t=a>0 ;

  • (9)

    there exists a positive constant C such that

    |f(t)|{C|t|,|t|1,C|t|12,|t|1;

  • (10)

    f(t)21/2pt for all t+.

Proposition 2.2.

The following properties involving f and h hold:

  • (1)

    (f(t))p-1f(t)t1-p is decreasing for t>0 ;

  • (2)

    (f(t))2r-1f(t)t1-p is increasing for rp and t>0 ;

  • (3)

    h(f(t))f(t)t1-p is increasing for t>0 ;

  • (4)

    F(t):=1p(f(t))2p*-1f(t)t-12p*f2p*(t) is increasing for t>0 ;

  • (5)

    1ph(f(t))f(t)t-H(f(t)) is increasing for t>0.

Proof.

(1) By computation, we have

ddt((f(t))p-1f(t)tp-1)=(p-1)(f(t)t)p-2ddt(f(t)t)f(t)+(f(t))p-1tp-1f′′(t)=(p-1)(f(t)t)p-2ddt(f(t)t)f(t)-2p-1(f(t))2p-2tp-1|f(t)|p+2.

Since f(t)t is a decreasing function, we can obtain ddt(f(t)t)<0. Thus,

ddt((f(t))p-1f(t)tp-1)<0for t>0.

(2) By Lemma 2.1 (4) and (7), we have that

ddt((f(t))2r-1f(t)tp-1)=(2r-1)f(t)2r-2(f(t))2tp-1+(f(t))2r-1f′′(t)tp-1-(p-1)(f(t))2r-1f(t)tp-2t2p-2=(f(t))2r-2f(t)tp[(2r-1)f(t)t-2p-1(f(t))p|f(t)|p+1t-(p-1)f(t)](f(t))2r-2f(t)tp(2r-2p)f(t)t0.

(3) Since

h(f(t))f(t)tp-1=h(f(t))f(t)2p-1f(t)2p-1f(t)tp-1,

by using conclusion (2) and (H3), property (3) is proved.

(4) By Lemma 2.1 (4) and (7), we have that

F(t)=2p*-1p(f(t))2p*-2(f(t))2t+1p(f(t))2p*-1f′′(t)t+1p(f(t))2p*-1f(t)-(f(t))2p*-1f(t)(f(t))2p*-2f(t)[2p*-1pf(t)t-2p-1p(f(t))p+1(f(t))pt+(1p-1)f(t)]p*-pp(f(t))2p*-1f(t)>0.

(5) From (H3) we obtain that h(s)s>(2p-1)h(s). Setting G(t)=1ph(f(t))f(t)t-H(f(t)) and using Lemma 2.1 (4) and (7), we deduce that

G(t)=1ph(f(t))(f(t))2t-2p-1ph(f(t))(f(t))p-1(f(t))p+2t+(1p-1)h(f(t))f(t)=1f(t)[1ph(f(t))f(t)(f(t))2t-2p-1ph(f(t))(f(t))p(f(t))p+2t+(1p-1)h(f(t))f(t)f(t)]>1f(t)[2p-1ph(f(t))(f(t))2t-2p-1ph(f(t))(f(t))p(f(t))p+2t+(1p-1)h(f(t))f(t)f(t)]h(f(t))f(t)f(t)[2p-1pf(t)t-1pf(t)t+(1p-1)f(t)]h(f(t))f(t)f(t)(p-1pf(t)+1-ppf(t))=0

as desired. ∎

Under the change of variables, we can rewrite the functional I defined by (2.1) in the following form:

J(v):=I(f(v))=εppN|v|pdx+1pNV(x)|f(v)|pdx-NH(f(v))dx-12p*N|f(v+)|2p*dx,

which is well defined on the Banach space

X={vW1,p(N):NV(x)|v|pdx<}

endowed with the norm

vX=(N(|v|p+V(x)|v|p)dx)1p.

In view of conditions (H0) and (H1), by the standard arguments, we conclude that JC1(X,) and

J(v),φ=εpN|v|p-2vφdx+NV(x)|f(v)|p-2f(v)f(v)φdxNh(f(v))f(v)φdx-N|f(v+)|2p*-2f(v+)f(v+)φdx

for all v,φX. Moreover, the critical points of J are the weak solutions of the Euler–Lagrange equation associated with the functional J given by

-εpΔpv+V(x)|f(v)|p-2f(v)f(v)=h(f(v))f(v)+|f(v)|2p*-1f(v).

We observe that if vXLloc(N) is a critical point of the functional J, then u=f(v)W1,p(N)Lloc(N) is a weak solution of problem (1.1), that is,

εpN[(1+2p-1|u|p)|u|p-2uφ+2p-1|u|p|u|p-2uφ]dx+NV(x)|u|p-2uφdx=N[h(u)+|u|2p-2u]φdx

for all φC0(N).

3 Autonomous problem

In this section, we will study the existence of a positive ground state solution for the following equation:

{-Δpv+μ|f(v)|p-2f(v)f(v)=h(f(v))f(v)+|f(v)|2p*-1f(v)in N,vW1,p(N),v(x)>0in N,(Qm)

where μ is an arbitrary positive constant and 2p<N. The functional μ associated to problem (Qm) is given by

μ(v)=1pN|v|pdx+μpN|f(v)|pdx-NH(f(v))dx-12p*N|f(v+)|2p*dx,

which is well defined on the Banach space Wμ endowed with the norm

vμ=(N(|v|p+μ|v|p)dx)1p.

From the hypotheses (H0)–(H1) it is easy to verify that JμC1(Wμ,) and

μ(v),φ=N|v|p-2vφdx+Nμ|f(v)|p-2f(v)f(v)φdx-Nh(f(v))f(v)φdx-N|f(v+)|2p*-2f(v+)f(v+)φdx,

for all v,φWμ. Moreover, the weak solution v of (Qm) corresponds to the critical point of the functional μ.

Let us denote the Nehari manifold associated to (Qm) by 𝒩μ, that is,

𝒩μ={vWμ:v0andμ(v),v=0}.

3.1 Mountain pass geometry

Theorem 3.1 ([36]).

Let E be a real Banach space and J:ER a functional of class C1. Let S be a closed subset of E which disconnects E into distinct connected components E1 and E2. Suppose further that J(0)=0 and that the following conditions hold:

  • (a)

    0E1 and there is α>0 such that J|Sα>0 ;

  • (b)

    there is eE2 such that J(e)<0.

Then J possesses a (C)c sequence with cα>0 given by

c=infγΓmax0t1J(γ(t)),

where

Γ={γC([0,1],E):γ(0)=0,J(γ(1))<0}.

Consider the set Sμ(ρ)={vWμ:Qμ(v)=ρp} and define

Qμ(v)=N(|v|p+μ|f(v)|p)dx.

Since Qμ(v) is continuous, Sμ(ρ) is a closed subset of Wμ and it disconnects this space.

Lemma 3.2.

Suppose that conditions (V) and (H0)(H1) are satisfied. Then there exist ρ0,δ0>0 such that

μ(v)δ0for all vSμ(ρ0).

Proof.

By the hypotheses (H0)–(H1), we have that

NH(f(v))dxεpN|f(v)|pdx+CεN|f(v)|2p*dx.(3.1)

By (3.1), Lemma 2.1 (10) and the Sobolev inequality, we have that

μ(v)=1pN(|v|p+μ|f(v)|p)dx-NH(f(v))dx-12p*N|f(v)|2p*dx1pN(|v|p+(μ-ε)|f(v)|p)dx-C0N|v|p*dxC1N(|v|p+μ|f(v)|p)dx-C2(N|v|pdx)p*pC1ρp-C2ρp*,

where C0,C1,C2>0 are positive constants. Thus we choose ρ=ρ0 sufficiently small and there exists δ0>0 such that μ(v)δ0>0 for all vSμ(ρ0). ∎

Similar to the proofs of [2, Lemma 3.2 and 3.3], we can deduce the following two Lemmas.

Lemma 3.3.

Suppose that (V), (H0)(H1) and (H4) hold. Then for each vWμ{0} the following limits hold:

  • (1)

    if v+0 , then μ(tv)- as t+ ;

  • (2)

    if v+=0 , then μ(tv)+ as t+.

Lemma 3.4.

For every vWμ{0} with v+0, there exists a unique tv>0 such that tvvNμ. Moreover, Iμ(tvv)=maxt0Iμ(tv).

From Lemmas 3.23.4 it follows that μ possesses the mountain pass geometry with

cμ=infγΓμmaxt[0,1]μ(γ(t)),

where

Γμ={γC([0,1],Wμ):γ(0)=0,μ(γ(1))<0},

and cμ can be characterized by the following identity:

cμ=infv𝒩μμ(v)=infvWμ{0}maxt0μ(tv).(3.2)

Therefore, by Theorem 3.1, there exists a (C)cμ sequence {vn}Wμ of μ, that is,

μ(vn)cμand(1+vnμ)μ(vn)0  as n.(3.3)

Now, we will give the detailed properties of the above (C)cμ sequence in the following two Lemmas.

Lemma 3.5.

Let {vn} be a (C)cμ sequence of Iμ. Then the following holds:

  • (1)

    {vn} is bounded in Wμ and there exists a vWμ such that vnv in Wμ ;

  • (2)

    Jμ(v)=0 ;

  • (3)

    vn0 for n.

Proof.

(1) By (3.3), (H2) and Lemma 2.1 (4), we have that

cμ+on(1)=μ(vn)-2θμ(vn),vn(1p-2θ)N|vn|pdx+μ(1p-2θ)N|f(vn)|pdx+(1θNh(f(vn))f(vn)dx-NH(f(vn)))dx+(1θ-12p*)N|f(vn+)|2p*dx(1p-2θ)N|vn|pdx+μ(1p-2θ)N|f(vn)|pdx,

which implies that

N|vn|pdx+μN|f(vn)|pdxC1.

On the other hand, using Lemma 2.1 (9) and the Sobolev inequality, we have that

N|vn|pdx={|vn|1}|vn|pdx+{|vn|1}|vn|pdx{|vn|1}|f(vn)|pdx+{|vn|1}|vn|pdxN|f(vn)|pdx+(N|vn|pdx)pp.

Therefore, {vn} is bounded in Wμ, and there exists a vWμ such that vnv in Wμ. Hence, up to a subsequence, there exists vWμ such that

{vnvin Wμ,vnvin Llocs(N) for 1s<p,vn(x)v(x)a.e. xN.(3.4)

Moreover, using [27, Theorem 1.6], we can get

vnva.e. in N.(3.5)

Indeed, by translation, equation (Qm) is reduced to

-Δpv+μ|v|p-2v=h(f(v))f(v)+|f(v)|2p*-1f(v)-μ|f(v)|p-1f(v)+μ|v|p-2v.

Let f~(x,v)=h(f(v))f(v)+|f(v)|2p*-1f(v)-μ|f(v)|p-1f(v)+μ|v|p-2v. By hypotheses (H0)–(H1) and using Lemma 2.1 (2), (6), (7), and (10), we have that

limt0f~(x,t)|t|p-2t=0,lim|t|+f~(x,t)|t|p*-2t=c0>0,

where c0>0 is a constant. Thus we have verified all conditions of [27, Theorem 1.6, Step 2], hence (3.5) follows.

(2) Since C0(N) is dense in Wμ, we only need to show Jμ(vn),φ=0 for all φC0(N). We observe that

μ(vn),φ-μ(v),φ=N(|vn|p-2vn-|v|p-2v)φdx+μN[|f(vn)|p-2f(vn)f(vn)-|f(v)|p-2f(v)f(v)]φdx   +N(h(f(v))f(v)-h(f(vn))f(vn))φdx+N[|f(v+)|2p*-1f(v+)-|f(vn+)|2p*-1f(vn+)]φdx,

thus we need to show that the following limits hold:

limnN(|vn|p-2vn-|v|p-2v)φdx=0,(3.6)limnN[|f(vn)|p-2f(vn)f(vn)-|f(v)|p-2f(v)f(v)]φdx=0,(3.7)limnN(h(f(v))f(v)-h(f(vn))f(vn))φdx=0,(3.8)limnN[|f(v+)|2p*-1f(v+)-|f(vn+)|2p*-1f(vn+)]φdx=0(3.9)

for all φC0(N).

By (3.5), it is easy to show that (3.6) holds by the weak convergence argument.

Let v~n=vn-v; next we show that (3.7)–(3.9) hold. Using Lemma 2.1 (2), (6), (7) and Young’s inequality, we deduce that

||f(vn)|m-2f(vn)f(vn)-|f(v)|m-2f(v)f(v)|=|01ddt[|f(v+tv~n)|m-2f(v+tv~n)f(v+tv~n)]dt|01[(m-1)|v~n||f(v+tv~n)|m-2(f(v+tv~n))2+|v~n||f(v+tv~n)|m-1|f′′(v+tv~n)|]dt01[(m-1)|v~n||v+tv~n|m-2+2p-1|v~n||f(v+tv~n)|p+m-2|f(v+tv~n)|p+2]dtC3(|v~n|m-1+|v~n||v|m-2)C4|v~n|m-1+C5|v|m-1,(3.10)

where pm<2p. By (3.4), we obtain

|f(vn)|m-2f(vn)f(vn)-|f(v)|m-2f(v)f(v)-C4|v~n|m-10a.e. in N.

By the Hölder inequality, we have

N[|f(vn)|m-2f(vn)f(vn)-|f(v)|m-2f(v)f(v)-C4|v~n|m-1]φdxC5N|v|m-1φdxC5(N|v|mdx)m-1m(N|φ|mdx)1m.

From the Lebesgue dominated convergence theorem it follows that

limnN[|f(vn)|m-2f(vn)f(vn)-|f(v)|m-2f(v)f(v)-C4|v~n|m-1]φdx=0.

By (3.4), we deduce that

limnN[|f(vn)|m-2f(vn)f(vn)-|f(v)|m-2f(v)f(v)]φdx=C5limnN|v~n|m-1φdx=0.

Take m=p and m=q; then (3.7) and (3.8) follows. In the case m=2p, using Lemma 2.1 (10), we can deduce that

||f(vn+)|2p-2f(vn+)f(vn+)-|f(v+)|2p-2f(v+)f(v+)|C6(v~n+)p-1+C7(v+)p-1.

Since (v~n+)p-10 in Lp/(p-1)(suppφ), so we get

limnN[|f(vn+)|2p-2f(vn+)f(vn+)-|f(v+)|2p-2f(v+)f(v+)]φdx=limnN(v~n+)p-1φdx=0.

Hence (3.9) is proven.

Consequently, we obtain

μ(vn),φ-μ(v),φ0as n.

Meanwhile, if {vn} is a bounded (C)cμ sequence of Jμ, then

μ(vn)(Wμ)=(1+vnμ)supφWμμ(vn),φ1+vnμ(1+vnμ)μ(vn)(Wμ)φμ0

for any φWμ. Thus μ(vn),φ=0, and as a result μ(v),φ=0 for any φC0(N).

(3) Since {vn} is bounded in Wμ, we get that {vn-} is bounded in Wμ, where vn-=max{-vn,0}. Using (H0), we have that

on(1)=μ(vn),-vn-N|vn-|pdx+μN|f(vn-)|p-2f(vn-)f(vn-)(vn-)dxN|vn-|pdx+μ2N|f(vn-)|pdx12(N|vn-|pdx+μN|f(vn-)|pdx).

Thus Qμ(vn-)0. Similar to the proof of [37, Proposition 2.4], we can deduce that

N|vn-|pdx+μN|f(vn-)|pdxC8vn-μ

for some C8>0 independent of n. Therefore, we get vn-=0 in Wμ. Hence, we get vn=vn++on(1) in Wμ. ∎

Lemma 3.6.

Let {vn} be a (C)cμ sequence of Iμ with cμ<12NSN/p. Then one of the following conclusions holds:

  • (1)

    Qμ(vn)0 ;

  • (2)

    there exist {yn}N and positive constants R, ξ such that

    limninfBR(yn)|vn|pdxξ.

Proof.

Assume that (2) dose not occur, that is, for all R>0 there holds

limninfBR(yn)|vn|pdx=0.

By the vanishing Lemma in [28], we can assume vn0 in Ls(N) for all s(p,p*). By (H0)–(H1) and Lemma 2.1 (6) and (10), we have

NH(f(vn))dx=Nh(f(vn))f(vn)dxN(ε|f(vn)|p+Cε|f(vn)|q+1)dxεN|vn|p+CεN|vn|q+12dx=on(1),

Next, since

f(vn)f(vn)μ2vnμ

and thus

Jμ(vn),f(vn)f(vn)=on(1),

that is,

N[1+2p-1fp(vn)1+2p-1fp(vn)]|vn|pdx+μN|f(vn)|pdx=Nh(f(vn))f(vn)dx+N|f(vn+)|2p*dx+on(1)=N|f(vn+)|2p*dx+on(1).(3.11)

Denote by l0 a number such that

N[1+2p-1fp(vn)1+2p-1fp(vn)]|vn|pdx+μN|f(vn)|pdxlandN|f(vn+)|2p*dxl.

Assume that l>0; by the definition of S=inf{N|u|pdx:up=1}, we have

SN|f2(vn+)|pdx(N|f2(vn+)|p*dx)pp*=N2p|f(vn+)|p1+2p-1|f(vn+)|p|vn+|pdx(N|f2(vn+)|p*dx)pp*N(1+2p-1|f(vn+)|p1+2p-1|f(vn+)|p)|vn+|pdx(N|f(vn+)|2p*dx)pp*l1-pp*as n,

thus, lSNp. Combining this with (3.11), we obtain that

cμ=limn{1pN[|vn|p+μfp(vn)]dx-12p*N|f(vn+)|2p*dx-NH(f(vn))dx}limn{12pN[(1+2p-1|f(vn)|p1+2p-1|f(vn)|p)|vn|p+μfp(vn)]dx-12p*N|f(vn+)|2p*dx}=(12p-12p*)l12NSNp,

which yields a contradiction because cμ<12NSN/p. Thus l=0. ∎

For the least energy level cμ, we have the following estimate.

Lemma 3.7.

For any μ>0, there exists v0Wμ{0} such that

maxt0μ(tv0)<12NSNp𝑎𝑛𝑑cμ=infv𝒩μμ(v)<12NSNp,

where S denotes the best constant for the embedding D1,p(RN)Lp*(RN).

Proof.

Define a functional Iμ:WμL(N) by

Iμ(u)=1pN(1+2p-1|u|p)|u|pdx+μpN|u|pdx-NH(u)dx-12p*N|u+|2p*dx.

By the equivalent characteristic of cμ (see (3.2)), we only need to prove that there exists 0v0WμL(N) such that

supt0Iμ(tv0)<12NSNp.

From Lemma 3.3 we know that Iμ(tv0)- as t+; then there exists some t*>0 such that Iμ(t*v0)<0. Define γ*(t):=f-1(tt*v0); by the definition of cμ, we have

cμ:=infγΓsupt[0,1]μ(γ(t))supt[0,1]μ(γ*(t))supt0Iμ(tv0)<12NSNp.

Fix ε>0 and define the function

uε(x)=ψ(x)(ε+|x|pp-1)N-p2p,vε(x)=uε(x)uε2Lp12,

where ψC0(N,[0,1]) is such that 0ψ1 if |x|<1 and ψ(x)=0 if |x|2. By [21, Lemma 4.1] we know that uε verifies the following estimates:

N|uε2|pdx=K1εp-Np+O(1),(N|uε2|pdx)pp=K2εp-Np+O(1),

and

N|uε2|tdx={K3εN(p-1)-t(N-p)p+O(1)if t>N(p-1)N-p,K3|lnε|+O(1)if t=N(p-1)N-p,O(1)if t<N(p-1)N-p,

where K1, K2, K3 are positive constants independent of ε and S=K1K2.

By computations, vε verifies

vεL2p=1,vε2Lpp=S+O(εN-pp),vεLpp=O(εN-p2p)(3.12)

and

N|vε|qdx={O(ε(N-p)q2p2)if q<2p-2NN-p,O(ε(N-p)q2p2|lnε|)if q=2p-2NN-p,O(εN(p-1)-q2(N-p)p+q(N-p)2p2)if q>2p-2NN-p.(3.13)

Obviously, vεWμL(N), and by (H4) we have

Iμ(tvε)=tppN(|vε|p+tp2|vε2|p)dx+tppNμ|vε|pdx-NH(tvε)dx-t2p*2p*N|vε|2p*dxtppN(|vε|p+μ|vε|p)dx+t2p2pN|vε2|pdx-CtσσN|vε|σdx-t2p*2p*=:gμ(t).

It is clear that limtgμ(t)=- and gμ(t)>0 when t is small; then supt0gμ(t) is attained at some tε>0. It follows that

0=gμ(tε)=tεp-1N(|vε|p+μ|vε|p)dx+tε2p-1N|vε2|pdx-Ctεσ-1N|vε|σdx-tε2p*-1=tεp-1[N(|vε|p+μ|vε|p)dx+tεpN|vε2|pdx-Ctεσ-pN|vε|σdx-tε2p*-p].

We have

N(|vε|p+μ|vε|p)dx+tεpN|vε2|pdx=Ctεσ-pN|vε|σdx+tε2p*-ptε2p*-p,

so tε is bounded from above by some T1>0. On the other hand,

N|vε2|pdxtε2p*-2p+Ctεσ-2pN|vε|σdx.

Since σ>2p, combining (3.12) with (3.13) and choosing ε small enough, we have tε2p*-2pS/2, so tε is bounded from below by some T2>0 independent of ε. Next, we define

eμ(t):=t2p2pN|vε2|pdx-t2p*2p*,

which attains its unique global maximum at

t0=(N|vε2|pdx)12p*-2p.

Thus, by (3.12) and (3.13), using the fact that σ>2pN/(N-p)-2NN-p, we have that

maxt0Iμ(tvε)gμ(tε)eμ(t0)+tεppN(|vε|p+μ|vε|p)dx-CtεσσN|vε|σdx(12p-12p*)(N|vε2|pdx)2p*2p*-2p+C1N(|vε|p+μ|vε|p)dx-C2N|vε|σdx=12NSNp+O(εN-pp)+O(εN-p2p)-O(εN(p-1)-σ2(N-p)p+σ(N-p)2p2)<12NSNp

for ε>0 small enough. The proof is completed. ∎

3.2 The existence of the ground state solution for (Qm)

Now, we are able to prove the existence of positive ground state solution for problem (Qm).

Theorem 3.8.

Suppose that conditions (H0)(H5) are satisfied. Problem (Qm) has a positive ground state solution vCloc1,α(RN)L(RN) satisfying v(x)0 as |x|.

Proof.

From Section 3.1, we know that μ satisfies the mountain pass geometry. There exists a (C)cμ sequence {vn}Wμ of μ, which satisfies

μ(vn)cμ<12NSNpand(1+vnμ)μ(vn)0  as n.

From Lemma 3.5, up to a subsequence, there is a vWμ such that vnv in Wμ with μ(v)=0. Without loss of generality, we can suppose that v0; otherwise, if Qμ(vn)0, using Lemma 2.1 (4), we have that

N|vn|pdx+μN|f(vn)|p-2f(vn)f(vn)vndxN|vn|pdx+μN|f(vn)|pdx0.

From μ(vn),vn0 we conclude that

N[h(f(vn))f(vn)vn+(f(vn+))2p*-1f(vn+)vn+]dx0.

Under the assumptions (H0), (H2) and Lemma 2.1 (4), we have

NH(f(vn))dx+12p*N|f(vn+)|2p*dx1θ{vn0}h(f(vn))f(vn)dx+12p*N|f(vn+)|2p*dxC1N[h(f(vn))f(vn)vn+|f(vn+)|2p*-1f(vn+)vn+]dx0.

Therefore, we conclude that

μ(vn)=1pQμ(vn)-N[H(f(vn))+12p*|f(vn+)|2p*]dx0,

which is a contradiction with μ(vn)cμ>0. Therefore, Qμ(vn)0. By Lemma 3.6, there exist {yn}N and positive constants R, ξ such that

limninfBR(yn)|vn|pdxξ.

Define v^n(x)=vn(x+yn). Then {v^n} is also a (C)cμ sequence of μ and satisfies

v^nv^ in Wμ,μ(v^)=0,v^nv^ in Lp(BR).

Then

BR(x)|v^|pdx=limnBR(x)|v^n|pdx=limnBR(yn)|vn|pdxξv^0,

so we can assume that v0. By Lemma 2.1 (4), we get

|f(vn)|p-|f(vn)|p-2f(vn)f(vn)vn|f(vn)|p-|f(vn)|p-2f2(vn)=0(3.14)

and

1p|f(vn+)|2p*-1f(vn+)vn+-12p*|f(vn+)|2p*(12p-12p*)|f(vn+)|2p*>0.

From (H2) it follows that

1ph(f(vn))f(vn)vn-H(f(vn))1ph(f(vn))f(vn)vn-1θh(f(vn))f(vn)(12p-1θ)h(f(vn))f(vn)>0.(3.15)

Combining (3.14)–(3.15) with Fatou’s Lemma, we obtain

cμ=limn[μ(vn)-1pμ(vn),vn]=limninfμpN[|f(vn)|p-|f(vn)|p-2f(vn)f(vn)vn]dx+limninfN[1ph(f(vn))f(vn)vn-H(f(vn))]dx+limninfN[1p|f(vn+)|2p*-1f(vn+)vn+-12p*|f(vn+)|2p*]dxμpN[|f(v)|p-|f(v)|p-2f(v)f(v)v]dx+N[1ph(f(v))f(v)v-H(f(v))]dx+N[1p|f(v+)|2p*-1f(v+)v+-12p*|f(v+)|2p*]dx=μ(v)-1pμ(v),v=μ(v).

Then v0 is a critical point of μ satisfying μ(v)cμ. On the other hand, v𝒩μ and cμ=inf𝒩μμ imply that μ(v)cμ, therefore μ(v)=cμ.

Now we show that v is nonnegative since

N|v|p-2vφdx+Nμ|f(v)|p-2f(v)f(v)φdx=Nh(f(v))f(v)φdx+N|f(v+)|2p*-2f(v+)f(v+)φdx(3.16)

for all v,φWμ. Let φ=-v-; then

012Qμ(v-)N|v-|pdx+Nμ|f(v-)|p-2f(v-)f(v-)(v-)dx=0

implies that v-=0, and thus v0.

Next, we will prove the L-estimate of v and that it decays to zero at infinity.

For any R>0, 0<rR/2, set ηC(N), 0η1, with η(x)=1 if |x|R and η(x)=0 if |x|R-r and |η|2/r. For l>0, let

vl(x)={v(x),vl,l,v>l,

and

zl=ηpvlp(β-1)vandωl=ηvlβ-1v,

with β>1 to be determined later. Taking zl as a test function, we get

Nηpvlp(β-1)|v|pdx=Nh(f(v))f(v)vηpvlp(β-1)dx+N|f(v)|2p*-1f(v)vηpvlp(β-1)dx-pNηp-1vlp(β-1)v|v|p-2vηdx-p(β-1)Nηpvlp(β-1)-1v|v|p-2vvldx-Nμ|f(v)|p-1f(v)vηpvlp(β-1)dx.

By (H0) and (H1), we see that for any τ>0 there exists Dτ>0 such that

h(f(t))τf(t)p-1+Dτ(f(t))2p*-1for all t0.

Choose τ sufficiently small; by Lemma 2.1 (10), we have that

Nηpvlp(β-1)|v|pdxC2N|f(v)|2p*-1f(v)vηpvlp(β-1)dx-pNηp-1vlp(β-1)v|v|p-2vηdxC3Nvp*ηpvlp(β-1)dx+pNηp-1vlp(β-1)v|v|p-1ηdx.

For every ϑ>0, by Young’s inequality, we have that

Nηpvlp(β-1)|v|pdxC3Nvp*ηpvlp(β-1)dx+pϑNηpvlp(β-1)|v|pdx+pCϑNvp|η|pvlp(β-1)dxC3Nvp*ηpvlp(β-1)dx+C4Nvp|η|pvlp(β-1)dx,(3.17)

where we have chosen ϑ>0 sufficiently small.

On the other hand, by the Sobolev inequality and the Hölder inequality, we have

ωlp*pC5N|ωl|pdx=C5N|(ηvlβ-1v)|pdxC5βp(Nvp|η|pvlp(β-1)dx+Nηpvlp(β-1)|v|pdx).(3.18)

Combining (3.17) with (3.18), we obtain

ωlp*pC6βp(Nvp|η|pvlp(β-1)dx+Nηpvlp(β-1)vp*dx).(3.19)

Let β=p*p; using the fact that R-rR2, we have that

ωlp*pC6βp(Nvp|η|pvlp*-pdx+{|x|R-r}ηpvlp*-pvp*dx)C6βpNvp|η|pvlp*-pdx+C6βp(N(vηvlp*-pp)p*dx)pp*(|x|>R/2vp*dx)p*-pp*.

From the definition of ωl it follows that

(N(vηvlp*-pp)p*dx)pp*C6βpNvp|η|pvlp*-pdx+C6βp(N(vηvlp*-pp)p*dx)pp*(|x|>R/2vp*dx)p*-pp*.

Since vLp(N), for R>0 sufficient large there holds

|x|R/2vp*dxϑ.

Therefore,

(|x|R(vvlp*-pp)p*dx)pp*(|x|R-r(ηvvlp*-pp)p*dx)pp*C7βpNvp|η|pvlp*-pdxC7βprpNvp*dxC8rp.

Using Fatou’s Lemma in the variable l, we get

|x|Rv(p*)2pdx<|x|R-rηp*v(p*)2pdx<C9rp*<+.(3.20)

Next, we note that if

β=p*(t-1)ptwitht=(p*)2p(p*-p),

then

β>1,t>1,ptt-1<p*,vLpβtt-1(|x|R-r).

By (3.19) and the Hölder inequality, we have that

ωlp*pC6βp(|x|R-rvpβ|η|pdx+|x|R-rηpvp*-p|v|pβdx)C6βp[(|x|R-rvpβtt-1dx)t-1t(R-r|x|R|η|ptdx)1t+(|x|R-r(ηpvp*-p)tdx)1t(|x|R-rvpβtt-1dx)t-1t]C6βp{[RN-(R-r)N]1trp(|x|R-rvpβtt-1dx)t-1t+(|x|R-r(ηp(p*-p)p*vp*-p)tdx)1t(|x|R-rvpβtt-1dx)t-1t},

where we have used pp(p-p)p. By (3.20), we deduce that

ωlp*pC10βp(1rp*t+RNtrp)(|x|R-rvpβtt-1dx)t-1t.

Using Fatou’s Lemma, we obtain

vp*β,(|x|R)pβC10βp(1rp*t+RNtrp)vpβtt-1β,(|x|>R-r)pβ.

If we take ψ=p*(t-1)/pt, s=pt/(t-1), then

vβψs,(|x|R)C1pββ1β(1rp*t+RNtrp)1pβvβs,(|x|>R-r).(3.21)

Setting β=ψm(m=1,2,), we obtain

vψm+1s,(|x|R)C101pψ-mψmψ-m(1rp*t+RNtrp)1pψmvψms,(|x|>R-r).(3.22)

Note that p>p*/t, p>N/t. Therefore, if we choose rm:=2-(m+1)R, then it follows from inequalities (3.21) and (3.22) that

vψm+1s,(|x|R)vψm+1s,(|x|R-rm+1)C101pψ-mψmψ-m(2p*(m+1)tRp*t+2p(m+1))1pψmvψms,(|x|>R-rm)C101pψ-mψmψ-m(2×2p(m+1))1pψmvψms,(|x|>R-rm)C101pi=1mψ-iψi=1miψ-iexp(i=1mln(2×2p(i+1))pψi)vψs,(|x|>R-r1)C11vp*,(|x|>R/2).

Letting m in the last inequality, we have

v,(|x|>R)C11vp*,(|x|>R/2).

Therefore, for any ϑ>0 there exists an R>0 such that v,(|x|>R)ϑ. Consequently, we conclude that lim|x|v(x)=0. ∎

4 The nonautonomous problem

In this section, we will study the following problem (which is equivalent to (Pe)), which can be obtained under the change of variable εz=x:

{-Δpv+V(εx)|f(v)|p-2f(v)f(v)=h(f(v))f(v)+|f(v)|2p*-1f(v)in N,vW1,p(N),v(x)>0in N.(Pe*)

The functional Jε corresponding to problem (Pe*) is given by

Jε(v)=1pN|v|pdx+1pNV(εx)|f(v)|pdx-NH(f(v))dx-12p*N|f(v+)|2p*dx,

which is well defined on the Banach space

Xε={vW1,p(N):NV(εx)|v|pdx<}

endowed with the norm

vXε=(N(|v|p+V(εx)|v|p)dx)1p.

Obviously, JεC1(Xε,) with

Jε(v),φ=N|v|p-2vφdx+NV(εx)|f(v)|p-2f(v)f(v)φdx-Nh(f(v))f(v)φdx-N|f(v+)|2p*-2f(v+)f(v+)φdx

for all v,φXε. Moreover, the weak solution v of (Pe*) corresponds to the critical point of the functional Jε. We define the Nehari manifold associated to (Pe*) by ε, that is,

ε={vXε:v0andJε(v),v=0}.

Set

Q(v)=N(|v|p+V(εx)|f(v)|p)dx.

We first show that the Nehari manifold ε is bounded from below.

Lemma 4.1.

There exists a constant C>0 such that vXεC>0 for all vMε.

Proof.

By Lemma 2.1 (10), the Hölder inequality and the Sobolev inequality, we have that

N|f(v)|q+1dx(N|f(v)|pdx)τ(q+1)p(N|f(v)|2pdx)(1-τ)(q+1)2p2(1-τ)(q+1)2p(N|f(v)|pdx)τ(q+1)p(N|v|pdx)(1-τ)(q+1)2p2(1-τ)(q+1)2pV0-τ(q+1)pS(1-τ)(q+1)2p(NV(εx)|f(v)|pdx)τ(q+1)p(N|v|pdx)(1-τ)(q+1)2pCQ(v)(1+τ)(q+1)2p,(4.1)

where τ(0,1) verifies

1q+1=τp+1-τ2p.

By Lemma 2.1 (7) and the Sobolev inequality, we have

Nf2p(v+)dxC(N|f2(v+)|pdx)ppC(N|v|pdx)ppCQ(v)pp.(4.2)

Thus, for vε, by (4.1), (4.2), Lemma 2.1 (4), and hypotheses (H0), (H1) and (H4), we deduce that

0=N(|v|p+V(εx)|f(v)|p-2f(v)f(v)v)dx-Nh(f(v))f(v)vdx-N|f(v+)|2p-1f(v+)v+dx12N(|v|p+V(εx)|f(v)|p)dx-N|f(v+)|2pdx-Nh(f(v))f(v)dx12N(|v|p+(1-2εV0)V(εx)|f(v)|p)dx-N|f(v+)|2pdx-CεN|f(v)|q+1dx14Q(v)-CQ(v)pp-CQ(v)(1+τ)(q+1)2p.

Since (1+τ)(q+1)2p>1, we have Q(v)C>0 for some C>0. This implies that

vXεQ(v)1pC>0

for all vε. ∎

It is easy to check, by arguing as in Section 3, that Jε exhibits the mountain pass geometry (Theorem 3.1) and there exists a (C)cε sequence {vn}Xε, for which we can assume vn0, such that vnv in Xε for some vXε and Jε(v)=0 (similar arguments as in Lemma 3.5, using hypothesis (V)). Moreover, from Lemma 3.7, there exists a v0WV{0} such that

maxt0Jε(tv0)<maxt0V(tv0)<12NSNp,

and thus

cε=infεJε=infvXε{0}maxt0Jε(tv)<12NSNp.

Similar to the proof Lemma 3.4, there is a unique tv>0 such that Jε(tvv)=maxt0Jε(tv).

Similar to the proof of Lemma 3.6, we can characterize the (C)cε sequence in the following Lemma.

Lemma 4.2.

Let {vn} be a (C)cε sequence of Jε with cε<12NSN/p. Then one of the following conclusions holds:

  • (1)

    Q(vn)0 ;

  • (2)

    there exist {yn}N and positive constants R, ξ such that

    limninfBR(yn)|vn|pdxξ.

Lemma 4.3.

Suppose that {vn} is a (C)cε sequence of Jε in Xε with cε<12NSN/p and vn0 in Xε. If Q(vn)0, then cεcV, where cV is the minimax level of JV.

Proof.

Let {tn}(0,) be a sequence such that {tnvn}𝒩V. We claim that limnsuptn1. Assume by contradiction that there exist δ>0 and a subsequence still denoted by {tn} such that tn1+δ for all n. Since {vn} is bounded in Xε, we may assume that vn0 for all n. From Jε(vn),vn=on(1) we get

N[|vn|p+V(εx)|f(vn)|p-1f(vn)vn]dx=Nh(f(vn))f(vn)vndx+N|f(vn)|2p*-1f(vn)vndx+on(1).(4.3)

Also since {tnvn}𝒩V, we get

N[tnp|vn|p+V|f(tnvn)|p-1f(tnvn)tnvn]dx=Nh(f(tnvn))f(tnvn)tnvndx+N|f(tnvn)|2p*-1f(tnvn)tnvndx.(4.4)

From (4.3) and (4.4) we have

N[h(f(tnvn))|f(tnvn)|2p-1|f(tnvn)|2p-1f(tnvn)(tnvn)p-1-h(f(vn))|f(vn)|2p-1|f(vn)|2p-1f(vn)vnp-1]vnpdx   +N[|f(tnvn)|2p*-1f(tnvn)(tnvn)p-1-|f(vn)|2p*-1f(vn)vnp-1]vnpdx=N(V-V(εx))|f(vn)|p-1f(vn)vndx   +NV[|f(tnvn)|p-1f(tnvn)(tnvn)p-1-|f(vn)|p-1f(vn)vnp-1]vnpdx+on(1).

Given ξ>0, by (V) there exists R=R(ξ)>0 such that V(εx)V-ξ for all |εx|R. Since vn0 in Xε, we have vn0 in Llocs(N) for s[1,p) and vn0 a.e. in N. Hence, we get

N(V-V(εx))|f(vn)|p-1f(vn)vndx=|εx|<R(V-V(εx))|f(vn)|p-1f(vn)vndx+|εx|R(V-V(εx))|f(vn)|p-1f(vn)vndxξC+2V|εx|<R|vn|pdx=ξC+on(1).

This together with Proposition 2.2 (1) and (2) and the boundedness of {vn} in Xε leads to

N[h(f(tnvn))|f(tnvn)|2p-1|f(tnvn)|2p-1f(tnvn)(tnvn)p-1-h(f(vn))|f(vn)|2p-1|f(vn)|2p-1f(vn)vnp-1]vnpdxξC+on(1).(4.5)

If Q(vn)0 in , by Lemma 4.2, there exist {yn}N and positive constants R, η such that

limninfBR(yn)|vn|pdxη.(4.6)

Define v~n=vn(x+yn); then there is a v~ such that, up to a subsequence, v~nv~ in Xε, v~nv~ in Ls(BR(0)), s[1,p), and v~nv~ a.e. in N. By (4.6), there exists a subset ΩBR(0) with a positive measure such that v~>0 a.e. in Ω. It follows from (4.5), (H3), Proposition 2.2 (2), and tn1+δ that

N[h(f((1+δ)v~n))|f((1+δ)v~n)|2p-1|f((1+δ)v~n)|2p-1f((1+δ)v~n)((1+δ)v~n)p-1-h(f(v~n))|f(v~n)|2p-1|f(v~n)|2p-1f(v~n)v~np-1]v~npdxξC+on(1).(4.7)

Let n in (4.7); using Fatou’s Lemma, we get

0<Ω[h(f((1+δ)v~))|f((1+δ)v~)|2p-1|f((1+δ)v~)|2p-1f((1+δ)v~)((1+δ)v~)p-1-h(f(v~))|f(v~)|2p-1|f(v~)|2p-1f(v~)v~p-1]v~pdxξC

for any ξ>0, which leads to a contradiction. Thus, limnsuptn1.

Next, we distinguish the following two cases. Case 1: limnsuptn=1. There exists a subsequence, still denoted by {tn}, such that tn1 as n. Hence

cε+on(1)=Jε(vn)cV+Jε(vn)-V(tnvn).(4.8)

By the boundedness of {vn} in Xε and (V), similar to the argument of (4.5), we have

Jε(vn)-V(tnvn)=1-tnppN|vn|pdx+1pN[V(εx)|f(vn)|p-V|f(tnvn)|p]dx+N[H(f(tnvn))-H(f(vn))]dx+12p*N[|f(tnvn)|2p*-|f(vn)|2p*]dxon(1)+1pNV[|f(vn)|p-|f(tnvn)|p]dx+N[H(f(tnvn))-H(f(vn))]dx+12p*N[|f(tnvn)|2p*-|f(vn)|2p*]dx-ξC.(4.9)

Using the mean value theorem and (H0)–(H1), we have

|N[H(f(tnvn))-H(f(vn))]dx|=|Nh(f(τvn))f(τvn)(tn-1)vndx|N(C1|vn|p-1+C2|vn|q+12)|tn-1|dx,

where τ is between 1 and tn. Using the Hölder inequality and limn(tn-1)=0, we obtain

N[H(f(tnvn))-H(f(vn))]dx=on(1),

NV[|f(vn)|p-|f(tnvn)|p]dx=on(1)

and

N[|f(tnvn)|2p*-|f(vn)|2p*]dx=on(1).

Combining (4.8) with (4.9), we obtain the following inequality:

cε+on(1)cV-Cξ+on(1).

Letting n in the above inequality, we have cεcV-Cξ for all ξ>0, thus cεcV.

Case 2: limnsuptn=t0<1. There exists a subsequence, still denoted by {tn}, such that tnt0 as n and tn<1 for all n. By Proposition 2.2 (1), (4) and (5), we see that

|f(t)|p-|f(t)|p-1f(t)t,1ph(f(t))f(t)t-H(f(t)),1p|f(t)|2p*-1f(t)t-12p*|f(t)|2p*

for t>0 are nondecreasing. Then

cVV(tnvn)-1pV(tnvn),tnvn=1pNV[|f(tnvn)|p-|f(tnvn)|p-1f(tnvn)tnvn]dx+N[1ph(f(tnvn))f(tnvn)tnvn-H(f(tnvn))]dx+N[1p|f(tnvn)|2p*-1f(tnvn)tnvn-12p*|f(tnvn)|2p*]dx1p{|εx|>R}(V(εx)+ξ)[|f(vn)|p-|f(vn)|p-1f(vn)vn]dx+1p{|εx|R}V[|f(vn)|p-|f(vn)|p-1f(vn)vn]dx+N[1ph(f(vn))f(vn)vn-H(f(vn))]dx+N[1p|f(vn)|2p*-1f(vn)vn-12p*|f(vn)|2p*]dx1pNV(εx)[|f(vn)|p-|f(vn)|p-1f(vn)vn]dx+N[1ph(f(vn))f(vn)vn-H(f(vn))]dx+N[1p|f(vn)|2p*-1f(vn)vn-12p*|f(vn)|2p*]dx+Cξ+on(1)=Jε(vn)-1pJε(vn),vn+Cξ+on(1)=cε+Cξ+on(1).

Letting ξ0, n0, we have cεcV. ∎

4.1 Compactness condition

Lemma 4.4.

Let {vn} be a (C)cε sequence of Jε in Xε and vnv in Xε for some vXε. Then

Jε(v~n)=Jε(vn)-Jε(v)+on(1),Jε(v~n)=on(1),

where v~n=vn-v.

Proof.

Firstly, we show that the following equalities hold:

NH(f(v~n))dx=NH(f(vn))dx+NH(f(v))dx+on(1),(4.10)NV(εx)|f(v~n)|pdx=NV(εx)|f(vn)|pdx+NV(εx)|f(v)|pdx+on(1),(4.11)NV(εx)[|f(v~n)|p-1f(v~n)-|f(vn)|p-1f(vn)+|f(v)|p-1f(v)]φdx=on(1),(4.12)N[h(f(v~n))f(v~n)-h(f(vn))f(vn)+h(f(v))f(v)]φdx=on(1),(4.13)Nf2p*(v~n+)dx=Nf2p*(vn+)dx-Nf2p*(v+)dx+on(1),(4.14)N(f2p*-1(v~n+)f(v~n+)-f2p*-1(vn+)f(vn+)+f2p*-1(v+)f(v+))φdx=on(1)(4.15)

for all φC0(N), where α(p,p*).The proof of (4.10)–(4.15) is similar to the proof of Lemma 3.5 (2). Here we only show that (4.14) holds true. Observe that by Lemma 2.1 (6), (7) and (10), we have that

|f2p*(vn+)-f2p*(v+)|=|01ddtf2p*((v~n+tv)+)dt|=|01[2p*f2p*-2((v~n+tv)+)f((v~n+tv)+)f((v~n+tv)+)v]dt|C01(|v~n|+|v|)p*-1|v|dtC(|v~n|p*-1|v|+|v|p*).

From this and Young’s inequality, for each δ>0 there exists Cδ>0 such that

|f2p*(v~n+)-f2p*(vn+)+f2p(v+)|δ|v~n|p*+Cδ|v|p*.

Define

Gδ,n(x)=max{|f2p*(v~n+)-f2p*(vn+)+f2p(v+)|-δ|v~n|p*,0},

which verifies that

Gδ,n(x)0 a.e. in N,0Gδ,n(x)Cδ|v|pL1(N).

Hence, by Lebesgue’s theorem, we have

NGδ,n(x)dx0as n.

By the definition of Gδ,n, we see that

|f2p*(v~n+)-f2p*(vn+)+f2p(v+)|δ|v~n|p*+CGδ,n(x).

Thus, we get

lim supnN|f2p*(v~n+)-f2p*(vn+)+f2p(v+)|dxCδ

which implies that

N|f2p*(v~n+)-f2p*(vn+)+f2p(v+)|dx=on(1)

and (4.14) follows.

Secondly, similar to the proof of (3.6) and from [10, Brezis–Lieb Lemma], we can deduce that

N|v~n|pdx=N|vn|pdx-N|v|pdx+on(1).(4.16)

Finally, by (4.10)–(4.16) and (3.6)–(3.9), we obtain

Jε(v~n)=Jε(vn)-Jε(v)+on(1)

and

Jε(v~n),φ=Jε(vn),φ-Jε(v),φ+on(1)=on(1)

for all φC0(N). ∎

Lemma 4.5.

The functional Jε satisfies the (C)cε condition at any level cε<cV.

Proof.

Let {vn} be a (C)cε sequence of Jε in Xε; then

Jε(vn)cε,(1+vnXε)Jε(vn)0.

By the boundedness of {vn} in Xε, we know that there exists vXε such that vnv in Xε and Jε(v)=0. Let v~n=vn-v; by Lemma 4.4, we have Jε(v~n)0 and

Jε(v~n)=Jε(vn)-Jε(v)+on(1)=cε-Jε(v)+on(1):=d+on(1).

From (H2) and Lemma 2.1 (4), we have

Jε(v)=Jε(v)-1pJε(v),v=1pNV(εx)[fp(v)-fp-1(v)f(v)v]dx+N[1ph(f(v))f(v)v-H(f(v))]dx+N[1p|f(v)|2p*-1f(v)v-12p*|f(v)|2p*]dx0.

Since V<, we have dcε<cV. It follows from Lemma 4.3 that Q(v~n)0. By [37, Proposition 2.4], we have v~n0 in Xε, that is, vnv in Xε. ∎

In order to apply the Ljusternik–Schnirelman category theory, we need the functional Jε to satisfy the compactness condition (such as (PS)c or (C)c condition) on the Nehari manifold. The following two Lemmas will explore this property.

Lemma 4.6.

The Nehari manifold Mε is of C1 class and ε(v),v<0 for any vMε, where ε:XεR is given by

ε(v)=N[|v|p+V(εx)|f(v)|p-2f(v)f(v)v]dx-Nh(f(v))f(v)vdx-N|f(v+)|2p*-1f(v+)vdx.

Proof.

Observe that

ε(v),v=pN|v|pdx+(p-1)NV(εx)|f(v)|p-2|f(v)|2v2dx+NV(εx)|f(v)|p-2f(v)f′′(v)v2dx+NV(εx)|f(v)|p-2f(v)f(v)vdx-Nh(f(v))|f(v)|2v2dx-Nh(f(v))f′′(v)v2dx-Nh(f(v))f(v)vdx-(2p*-1)N|f(v+)|2p*-2|f(v+)|2v2dx-N|f(v+)|2p*-1f′′(v+)v2dx-N|f(v+)|2p*-1f(v+)vdx.

By vε, we deduce that

ε(v),v=NV(εx)((p-1)|f(v)|p-2|f(v)|2v2+V(εx)|f(v)|p-2f(v)f′′(v)v2-(p-1)V(εx)|f(v)|p-2f(v)f(v)v)dx-Nh(f(v))|f(v)|2v2dx-Nh(f(v))f′′(v)v2dx+(p-1)Nh(f(v))f(v)vdx-(2p*-1)N|f(v+)|2p*-2|f(v+)|2v2dx-N|f(v+)|2p*-1f′′(v+)v2dx+(p-1)N|f(v+)|2p*-1f(v+)vdx.

Let

g~(t)=(p-1)|f(t)|p-2|f(t)|2t2-2p-1|f(t)|p-2|f(t)|p|f(t)|p+2t2-(p-1)|f(t)|p-2f(t)f(t)t

for t; according to the definition of f, we obtain that g~(t)=g~(-t) for t. Note that by Lemma 2.1 (4), we have that g~(t)0 for t0. Thus g~(t)0 for all t. Hence, we get

NV(εx)((p-1)|f(v)|p-2|f(v)|2v2+|f(v)|p-2f(v)f′′(v)v2-(p-1)|f(v)|p-2f(v)f(v)v)dx=NV(εx)((p-1)|f(v)|p-2|f(v)|2v2-2p-1|f(v)|2p-2|f(v)|p+2v2-(p-1)|f(v)|p-2f(v)f(v)v)dx=NV(εx)g~(v)dx0.(4.17)

From Proposition 2.2 (3) and h(s)=0 for s0, we have

h(f(s))|f(s)|2v2+h(f(s))f′′(s)s2-(p-1)h(f(s))f(s)s0for all s.

Thus

N[h(f(v))|f(v)|2v2+h(f(v))f′′(v)v2-(p-1)h(f(v))f(v)v]dx0.(4.18)

Therefore, by (4.17), (4.18) and Lemma 2.1 (4) and (7), we have that

ε(v),v-(2p*-1)N|f(v+)|2p*-2|f(v+)|2v2dx-N|f(v+)|2p*-1f′′(v+)v2dx+(p-1)N|f(v+)|2p*-1f(v+)vdx-(2p*-1)N|f(v+)|2p*-2|f(v+)|2v2dx+2p-1N|f(v+)|2p*-1|f(v+)|p-1|f(v+)|p+2v2dx+2(p-1)N|f(v+)|2p*-2|f(v+)|2v2dx2(p-p*)N|f(v+)|2p*dx<0.(4.19)

Lemma 4.7.

The functional Jε restricted to Mε satisfies the (C)cε condition at any level cε<cV.

Proof.

Let J~ε=Jε|ε. Let {vn}ε such that J~ε(vn)cε, (1+vnXε)J~ε(vn)0. Thus, using vnε and J~ε(vn)=Jε(vn)c, similar to the proof of Lemma 3.5, we conclude that {vn} is bounded in Xε. By Lemma 4.5, the constrained gradient has the form

J~ε(v)=Jε(v)-Jε(v),ε(v)ε(v)2ε(v).

For vnε being a (C)cε sequence, we denote

λn=Jε(vn),ε(vn)ε(vn)2,

and then we have that

(1+vnXε)Jε(vn)=λn(1+vnXε)ε(vn)+on(1).

By (4.19), we see that ε(vn),vnγ0; if γ=0, we have f(vn+)0 in L2p*(N). Therefore, using an interpolation argument, the boundedness of {vn} in Xε and (H0)–(H1), we deduce that

12N(|vn|p+V(εx)|f(vn)|p)dxN(|vn|p+V(εx)|f(vn)|p-2f(vn)f(vn)vn)dx=Nh(f(vn+))f(vn+)vn+dx+N|f(vn)|2p*-1f(vn)vndxNh(f(vn+))f(vn+)dx+N|f(vn+)|2p*dxεN|f(vn+)|pdx+CεN|f(vn+)|q+1dx+N|f(vn+)|2p*dx0

as n, which contradicts Lemma 4.1. Thus γ0. This together with vnε leads to

0=(1+vnXε)Jε(vn),vn=λn(1+vnXε)ε(vn),vn+on(1)(1+vnXε),

and so λn=on(1). Thus (1+vnXε)Jε(vn)=on(1). We have proved that {vn} is a (C)cε sequence of Jε in Xε; the conclusion is obtained by Lemma 4.5. ∎

By a similar argument, or using Lemma 4.6, we get the following corollary.

Corollary 4.8.

The critical points of Jε on Mε are critical points of Jε in Xε.

4.2 The existence of the ground state solution for (Pe*)

Theorem 4.9.

Suppose that conditions (V) and (H0)(H5) are satisfied. Then there exists ε¯>0 such that problem (Pe*) has a ground state solution uεCloc1,α(RN)L(RN) for all 0<ε<ε¯.

Proof.

From the above statement, we know that Jε satisfies the mountain pass geometry. By Theorem 3.1, there exists a (C)cε sequence {vn}Xε of Jε satisfying

Jε(vn)cεand(1+vnXε)Jε(vn)0  as n.

Without loss of generalization, we may assume that V0=V(0)=infxNV(x). Let μ such that V0<μ<V; we have that cV0<cμ<cV. Let ωμ be a nonnegative ground state of problem (Qm) and let ϕC0(N,[0,1]) be such that ϕ(x)=1 for |x|1 and ϕ(x)=0 for |x|2. For R>0, set ϕR(x)=ϕ(x/R), and let uR=ϕR(x)ωμ. By Lemma 3.4, there exists tR>0 such that vR=tRuR𝒩μ. Then there exists R0>0 such that vR0𝒩μ satisfies μ(vR0)<cV. If not, μ(vR)cV for all R>0; by ωμ𝒩μ and uRωμ in Wμ as R, we obtain that tR1. Thus

cVlim infRμ(tRuR)=μ(ωμ)=cμ<cV.

This achieves a contradiction. Since suppvR0 is a compact set, we may choose ε¯>0 such that V(εx)μ for any ε(0,ε¯) and xsuppvR0. Thus

NV(εx)|f(vR0)|pdxNμ|f(vR0)|pdxfor all ε(0,ε¯).

Therefore, for all ε(0,ε¯) and t0, we have that

Jε(tvR0)μ(tvR0)μ(vR0)<cV,

which implies that cε<cV for all ε(0,ε¯).

By Lemma 4.5, there exists a vXε (the limit of {vn}) such that

Jε(v)=cεandJε(v)=0.

That is, vXε is a solution of problem (Pe*). By a standard argument, we can obtain that vCloc1,α(N) with 0<α<1 and vL(N). ∎

4.3 Multiplicity of solutions to (Pe*)

In this subsection, we will study the multiplicity of solutions and study the behavior of its maximum points concentrating on the set M of global minima of V given in Section 1. The main result of this section is equivalent to Theorem 1.1 and it can be restated as follows.

Theorem 4.10.

Suppose that conditions (V) and (H0)(H5) are satisfied. For a given δ>0 there exists εδ>0 such that for any ε(0,εδ) problem (Pε*) has at least catMδ(M) positive weak solutions in Cloc1,α(RN)L(RN). Moreover, each solution decays to zero at infinity and if uε denotes one of these positive solutions and zεRN its global maximum, then

limε0V(εzε)=V0.

To prove Theorem 4.10, we fix some notation and give some preliminary lemmas. Fix δ>0 and let ω be a ground state solution of problem (𝒬V0). Let η be a smooth nonincreasing cut-off function defined in [0,) such that η(s)=1 if 0sδ2 and η(s)=0 if sδ. For any ε>0 and yM, define a function ψε,y(x) by

ψε,y(x)=η(|εx-y|)ω(εx-yε),

tε>0 satisfying

maxt0Jε(tψε,y)=Jε(tεψε,y)

and ϕε:Mε by

ϕε(y)=tεψε,y.

By construction, ϕε(y) has compact support for any yM.

For any δ>0, let ρ=ρ(δ)>0 be such that MδBρ(0). Let χ:NN be defined as χ(x)=x for |x|ρ and χ(x)=ρx|x| for |x|ρ. Finally, let us consider β:εN given by

β(u)=Nχ(εx)|u(x)|pdxN|u(x)|pdx.

Lemma 4.11.

The function ϕε satisfies the following limit:

limε0Jε(ϕε(y))=cV0uniformly in yM,

and

limε0β(ϕε(y))=yuniformly in yM.

Proof.

The proof of this lemma can be found in [2]; we omit its proof. ∎

Lemma 4.12 (A compactness lemma).

Let {vn}Nμ be a sequence satisfying Iμ(vn)cμ. Then one of the following holds:

  • (1)

    {vn} has a subsequence strongly convergent in Wμ ;

  • (2)

    there exists a sequence {yn}N such that, up to a subsequence, v^n=vn(x+yn) converges strongly in Wμ . In particular, there exists a minimizer for cμ.

Proof.

Since {vn}𝒩μ and μ(vn)cμ, it is easy to check that {vn} is bounded in Wμ. Since cμ=infv𝒩μμ, we can use the Ekeland variational principle (see [39, p. 122, Theorem 8.5]): there exists ωn𝒩μ such that ωn=vn+on(1), μ(ωn)cμ and

μ(ωn)=λnμ(ωn)+on(1).

Using the boundedness of {vn}, we obtain that

(1+ωnμ)μ(ωn)=λn(1+ωnμ)μ(ωn)+on(1),

where λn is a real number and μ(ω)=μ(ω),ω for any ωWμ. We claim that there exists α0>0 such that |μ(ωn),ωn|α0 for all n. Indeed, using a similar argument as we have done in Lemmas 4.6 and 4.7, we have λn=on(1), which yields

ωn=vn+on(1),μ(ωn)cμ,(1+ωnμ)μ(ωn)0.

So without loss of generality, we may suppose that {vn} is a (C)cμ for μ. Hence, up to a subsequence still denoted by {vn}, we may assume that there exists vWμ such that vnv in Wμ, vnv in Llocs(N) for s[1,p), vn(x)v(x) a.e. in N (see (3.5) in Lemma 3.5), and vnv a.e. in N. Moreover, from Lemma 3.5, we see that μ(v)=0 and vn0 for all n.

Case 1: v0. In this case, from the semi-continuity of the semi-norm, we have

N|v|pdxlimninfN|vn|pdx.

We claim that equality holds in the last inequality. Otherwise, using the fact that

1p|f(t)|p-2θ|f(t)|p-1f(t)t,2θh(f(t))f(t)t-H(f(t)),2θ|f(t)|2p*-1f(t)t-12p*|f(t)|2p*

are nonnegative functions for t0, and by Fatou’s Lemma, we have that

cμμ(v)=μ(v)-2θμ(v),v=(1p-2θ)N|v|pdx+N[μp|f(v)|p-2θμ|f(v)|p-1f(v)v]dx+N[2θh(f(v))f(v)v-H(f(v))]dx+N[2θ|f(v)|2p*-1f(v)v-12p*|f(v)|2p*]dx<limn{(1p-2θ)N|vn|p+N[μp|f(vn)|p-2θμ|f(vn)|p-1f(vn)vn]dx+N[2θh(f(vn))f(vn)vn-H(f(vn))]dx+N[2θ|f(vn)|2p*-1f(vn)vn-12p*|f(vn)|2p*]dx}=limn(μ(vn)-2θμ(vn),vn)cμ,

which leads to a contradiction. So we obtain

N|v|pdx=limnN|vn|pdx.

Combining (4.20) with vn(x)v(x) a.e. in N and the Brezis–Lieb Lemma [10], we can conclude that

N|vn-v|pdx0as n.(4.20)

By an argument similar to (4.20), we also have that

limnN[μp|f(vn)|p-2θμ|f(vn)|p-1f(vn)vn]dx=N[μp|f(v)|p-2θμ|f(v)|p-1f(v)v]dx.

Hence, up to a subsequence still denoted by {μp|f(vn)|p-2θμ|f(vn)|p-1f(vn)vn}, there exists k(x)L1(N) such that

θ-2ppθμ|f(vn)|pμp|f(vn)|p-2θμ|f(vn)|p-1f(vn)vnk(x)a.e. in N,

where we have used Lemma 2.1 (4) and the fact that vn0. By the Lebesgue dominated convergence theorem, we have that

limnNμ|f(vn)|pdx=Nμ|f(v)|pdx,

which implies that

limnNμ|f(vn)-f(v)|pdx=0.

Hence, up to a subsequence, there exists k1(x)L1(N) such that

μ(|f(vn)|p-|f(v)|p)k1(x)a.e. in N.

Using the convexity of |f|p and the evenness of |f|p, we have that

V(x)|f(vn-v)|pV(x)2|f(2vn)|p+V(x)2|f(2v)|p2p-1(V(x)|f(vn)|p+V(x)|f(v)|p)2p(k1(x)+V(x)|f(v)|p).

Since k1(x)+V(x)|f(v)|pL1(N), by the condition vnv a.e. in N and the Lebesgue dominated convergence theorem, we get

limnNμ|f(vn-v)|pdx=0.(4.21)

Therefore, by Lemma 2.1 (9), the Sobolev inequality, (4.20), and (4.21), we deduce that

Nμ|vn-v|pdx={|vn-v|1}μ|vn-v|pdx+{|vn-v|1}μ|vn-v|pdxC{|vn-v|1}μ|f(vn-v)|pdx+{|vn-v|1}μ|vn-v|pdxCNμ|f(vn-v)|pdx+μS1pN|vn-v|pdx0as n.

Consequently, we conclude that vnv in Wμ.

Case 2: v0. By Lemma 3.6, there exist a sequence R, a>0 and {yn}N such that

BR(yn)|vn|pdxa>0.

Let v^n=vn(x+yn); then μ(v^n)cμ and (1+v^nμ)μ(v^n)0 as n. It is clear that there exists v^Wμ such that v^nv^ in Wμ. Then we use the discussion given in Case 1 to obtain the result. ∎

Lemma 4.13.

Let εn0+ and (vn)Mεn be such that Jεn(vn)cV0. Then there exists a sequence (yn~)RN such that v^n=vn(x+yn~) has a convergent subsequence in WV0. In particular, up to a subsequence, we have εnyn~yM.

Proof.

In view of Jεn(vn),vn=0 and Jεn(vn)cV0, by an argument similar to Lemma 3.5 (1), we conclude that {vn} is bounded in WV0. Since cμ>0, we have QV0(vn)0 for all n. If not, it is easy to check that cμ0, a contradiction. By Lemma 3.6, there exist a sequence {yn~}N and positive constants R~, ξ such that

limninfBR(yn~)|vn|pdxξ>0.

Let v~n=vn(x+yn~); up to a subsequence, there exists v~WV0 such that v~nv~0 in WV0. Let tn>0 be such that ωn:=tnv~n𝒩V0. Using vnεn, we deduce that

cV0V0(ωn)1pN[|ωn|p+V(εnx+yn~)|f(ωn)|p]dx-12p*N|f(ωn+)|2p*dx-NH(f(ωn))dx=1pN[|(tnv~n)|p+V(εnx+yn~)|f(tnv~n)|p]dx-12p*N|f(tnv~n+)|2p*dx-NH(f(tnv~n))dx=Jεn(tnvn)Jεn(vn)=cV0+on(1).

Hence, limnV0(ωn)=cV0. From ωn𝒩V0 it follows that {ωn} is bounded in WV0. By the boundedness of {v~n}, we know that {tn} is bounded. Thus, up to a subsequence still denoted by {tn}, we may assume that tnt0. If t0=0, from the boundedness of v~n we have that ωn=tnvn~0. Hence, V0(ωn)0, which contradicts cV0>0. Thus t0>0. From the boundedness of {ωn} and v~nv~, up to a subsequence, we have that ωnω=t0v~ in WV0. By t0>0 and v~0, we see that ω0. From Lemma 4.12 we have ωnω in WV0, which implies that v~nv~ in WV0.

It remains to show that εnyn~ is bounded. In fact, suppose by contradiction that |εnyn~|. Since ωnω in WV0 and V0<V, it follows that

cV0=V0(ω)<V(ω)lim infn[1pN|ωn|pdx+1pNV(εnx+εnyn~)|f(ωn)|pdx-12p*N|f(ωn)|2p*dx-NH(f(ωn))dx]=lim infnJεn(tnvn)lim infnJεn(vn)=cV0,

which gives a contradiction. Thus, εnyn~ is bounded in N and, up to a subsequence, εnyn~y in N. If yM, then V(y)>V0 and we can obtain a contradiction by arguing as above. Hence yM. ∎

Let g:++ be a positive function such that g(ε)0 as ε0 and define the set

ε~={vε:Jε(v)cV0+g(ε)}.

Lemma 4.14.

Let δ>0; there holds that

limε0supvε~infyMδ|β(v)-y|=0.

Proof.

Let {εn}+ be such that εn0. For each n, there exists {vn}εn~ satisfying

infyMδ|β(vn)-y|=supv𝒩~εninfyMδ|β(v)-y|+on(1).

Thus it suffices to find a sequence {yn}Mδ such that

limn|β(vn)-yn|=0.

To obtain this sequence, we note that V0(tvn)Jε(tvn) for t0 and {vn}εn~εn, and so

cV0cεnJεn(vn)cV0+g(εn).

This implies that Jε(vn)cV0. By Lemma 4.13, we obtain a sequence {yn~}N such that εnyn~Mδ for n sufficiently large. Then

β(vn)=εnyn~+N[χ(εnz+εnyn~)-εnyn~]|vn(z+yn~)|pdxN|vn(z+yn~)|pdx.

Recalling that εnx+εnyn~yM, we have that β(vn)=εnyn~+on(1), and therefore the sequence {yn:=εnyn~} is required. ∎

The following Lemma plays a fundamental role in the study of the behavior of the maximum points of the solutions.

Lemma 4.15.

Suppose that conditions (V) and (H0)(H5) are satisfied. Let vn be a solution of the following problem:

{-Δpvn+Vn(x)|f(vn)|p-2f(vn)f(vn)=h(f(vn))f(vn)+|f(vn)|2p*-1f(vn)in N,vnW1,p(N),vn(x)>0in N,

where Vn(x)=V(εnx+εnyn~). If vnv in W1,p(RN) with v0, then vnL(RN) and vnL(RN)C for all nN. Moreover, lim|x|vn(x)=0 uniformly in n.

Proof.

We only replace v by vn and apply the fact that vnv in W1,p(N) in Theorem 3.8. ∎

Lemma 4.16.

Under the hypotheses of Lemma 4.15, there exists δ0>0 such that vnL(RN)δ0 for all nN.

Proof.

Suppose by contradiction that vnL(N)0 as n. Given ε0=V04, by (H1) there exists n0 such that

h(f(vn(x)))(f(vn(x)))p-1<ε0a.e. in N for nn0.

Thus, by Lemma 2.1 (4) and (9), for n large enough, we have

N(|vn|p+V02|vn|p)dxN(|vn|p+V02|f(vn)|p)dxN(|vn|p+Vn(x)|f(vn)|p-2f(vn)f(vn)vn)dx=N|f(vn)|2p*-1f(vn)vndx+Nh(f(vn(x)))(f(vn(x)))p-1(f(vn(x)))p-1f(vn)vndxε0N|f(vn)|pdx+N|f(vn)|2p*dx,

which implies that vn0 in W1,p(N), which contradicts the hypothesis that vnv with v0. Hence, there exists δ0>0 such that vnL(N)δ0 for all n. ∎

4.4 Proof of Theorem 4.10

Proof.

For a fixed δ>0, by Lemma 4.11 and Lemma 4.14, there exists εδ>0 such that for any ε(0,εδ), βϕε is well defined. Fix ε>0 small enough, so βϕε is homotopic to the inclusion map id:MMδ and by arguments similar to the ones contained in the proofs of [7, Lemmas 4.2 and 4.3], we obtain that

catε~(ε~)catMδ(M).

Since Jε satisfies the (C)cε condition for cε(cV0,cV0+g(ε)), using the Ljusternik–Schnirelman theory of critical points in [39] (see [39, Theorem 5.19]; it can be true under the (C)c condition), we know that Jε possesses at least catMδ(M) critical points on ε. Consequently, by Corollary 4.8, Jε has at least catMδ(M) critical points in Xε.

The remaining proof of concentration behavior can be deduced by using Lemma 4.15 and Lemma 4.16 and its proof is a standard argument; we refer the interested readers to [1, 2, 24]. We omit its details here. ∎

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About the article

Received: 2016-10-10

Revised: 2017-02-07

Accepted: 2017-02-25

Published Online: 2017-04-19


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11501403

Funding Source: Natural Science Foundation of Shanxi Province

Award identifier / Grant number: 2013021001-3

This work is supported by the NSFC (No. 11501403) and the Natural Science Foundation of Shanxi Province for Youths (No. 2013021001-3).


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 339–371, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0218.

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