Infinitely many solutions to quasilinear Schrödinger equations with critical exponent

Infinitely many solutions to quasilinear Schrödinger equations with critical exponent

Li Wang

, Jixiu Wang

and Xiongzheng Li

1School of Basic Science, East China Jiaotong University, Nanchang, 330013, China

2School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang 441053, China Received 2 July 2018, appeared 17 January 2019

Communicated by Dimitri Mugnai

Abstract.This paper is concerned with the following quasilinear Schrödinger equations with critical exponent:

−_∆pu+V(x)|u|^p−2u−_∆p(|u|^2ω)|u|^2ω−2u=ak(x)|u|^q−2u+b|u|^2ωp∗−2u, x∈_R^N. Here∆pu=_div(|∇u|^p−2∇u)_{is the}p-Laplacian operator with 1< _p<_N,p^∗= _N−p^{N p} _is the critical Sobolev exponent. 1≤2ω<q<2ωp,aandbare suitable positive parame- ters,V∈C(_R^N,[0,∞)),k∈C(_R^N,R). With the help of the concentration-compactness principle and R. Kajikiya’s new version of symmetric Mountain Pass Lemma, we obtain infinitely many solutions which tend to zero under mild assumptions onVandk.

Keywords: critical exponent, concentration-compactness principle, symmetric Moun- tain Pass Theorem.

2010 Mathematics Subject Classification: 35A15, 35J10, 35J62.

1 Introduction and main result

In this paper, we establish the existence of infinitely many solutions which tend to zero for the following quasilinear Schrödinger equations with critical exponent

−_∆pu+V(x)|u|^p−2u−_∆p(|u|^2ω)|u|^2ω−2u=ak(x)|u|^q−2u+b|u|^2ωp∗−2u, x∈_R^N_{. (1.1)} The energy functional associated with (1.1) is given by

I(u) = ¹ p

Z

R^N

|∇u|^p+V(x)|u|^pdx+ (_2ω)^p−1 p

Z

R^N|u|^p(2ω−1)|∇u|^pdx

− ^a q

Z

R^Nk(x)|u|^qdx− ^b 2ωp^∗

Z

R^N|u|^2ωp∗dx.

(1.2)

Here ∆pu = div(|∇u|^p−2∇u) is the p-Laplacian operator with 1 < p < N, p^∗ = _N^{N p}_−p is the critical Sobolev exponent. 1≤2ω<q<2ωp,aandbare positive parameters. V(x)andk(x) are continuous and satisfy the following conditions:

BCorresponding author. Emails: wangli.423@163.com (L. Wang), wangjixiu127@aliyun.com (J. Wang), 944595309@qq.com (X. Li).

(V) V ∈C(_R^N,[0,∞))satisfies inf_x∈RNV(x)≥ V₀> 0, and for each M> 0, meas{x∈_R^N : V(x) ≤ M} < +_{∞, where} V0 is a constant and meas denotes the Lebesgue measure inR^N.

(_K) 0<k(x)∈ L^r(_R^N)withr= _2ωp^2ωp∗−^∗q.

In recent years, a great attention has been focused on the study of solutions to quasilinear Schrödinger equations. Such equations arise in various branches of mathematical physics. For example, whenp=2,ω =1, the solutions of (1.1) are related to the existence of solitary wave solutions for quasilinear Schrödinger equations

ih∂Ψ

∂t =−_∆Ψ+W(x)_Ψ−h^˜(|_Ψ|²)_Ψ−κ∆[ρ(|_Ψ|²)]ρ⁰(|_Ψ|²)_Ψ, (1.3) where Ψ : R×_R^N → C,W : R^N → _R is a given potential, κ,h are real constants and ρ, ˜h are real functions. This type of equations appear more naturally in mathematical physics and have been derived as models of several physical phenomena corresponding to various types ofρ(s). In the caseρ(s) =s, (1.3) was used for the superfluid film equation in plasma physics by Kurihara in [12] and [13]. In the case ρ(s) = (1+s)^1/2, (1.3) models the self-channeling of a high-power ultrashort laser in matter (see [4,6]). Considering the case ρ(s) = s^α,κ > 0 and puttingΨ(t,x) =exp(−^iFt_h )u(x),F∈_Ris some real constant, it is clear thatΨ(t,x)solves (1.3) if and only ifu(x)solves the following elliptic equation:

−_∆u+V(x)u−ακ∆(|u|^2α)|u|^2α−2u=h^˜(x,u), x∈_R^N, (1.4) where we have renamedW(x)−F to beV(x).

For the caseακ=1, ˜h(x,u) =θ|u|^p−1u, Poppenberg, Schmitt and Wang in [19] studied the equation (1.4) by translating it into an ODE

−u⁰⁰+V(x)u−(u²)⁰⁰u=θ|u|^p−1u, x∈_R, (1.5) and then a ground state solutionu∈W^1,2(_R)of problem (1.5) was obtained. They also got that the equation (1.5) admits a positive solutionu∈W^1,2(_R)for any arbitrarily large values of θ.

Later, Liu, Wang and Wang in [17] established the existence of ground states of soliton-type solutions for (1.4) as in the case α= 1,κ= ¹₂ by the variational methods. Using a constrained minimization argument, Liu, Wang and Wang in [16] established the existence of a positive ground state solution for (1.4). As we know, Nehari method is used to get the existence results of ground state solutions in [10] and the problem is transformed to a semilinear one in [2,9] by a change of variables. Recently, the author in [23] studied the equation (1.4) and obtained that it has a positive and a negative weak solution under proper conditions of α,V,g. A natural question is that weather there exist infinitely many solutions for equations like (1.4). The authors in [7,8] investigated the following type quasilinear elliptic equation:

−_∆u+V(x)u−_∆I(u²)I⁰(u²)u=h(u), x∈_R^N. (1.6) Let

g²(u) =1+[(I(u²))⁰]²

2 .

Problem (1.6) can be reduced to the following quasilinear elliptic equations:

−_div(g²(u)∇u) +g(u)g⁰(u)|∇u|²+V(x)u=h(u)_, x∈_R^N_{. (1.7)}

By using the Pohozaev identity, the author has the nonexistence result for (1.7).

To the best of our knowledge, the existence of nontrivial radial solutions for (1.4) with g(x,u) = µu^2(2∗)−1 was firstly studied by Moameni in [18], where the Orlicz space as the same as it was used in [17]. However, it seems that there is almost no work on the existence of infinitely many solutions to the quasilinear Schrödinger problem in R^N involving critical nonlinearities and generalized potentialV(x)_.

Motivated by the above discussions, the main goal of this paper is to study the existence of infinitely many solutions which tend to zero to the problem (1.1). The lack of compactness of the embedding from W^1,p(_R^N)into L^p∗(_R^N)prevents us from using the variational methods in a standard way. To overcome the lack of compactness caused by the Sobolev embeddings in unbounded domains and the critical exponent, some new estimates for (1.1) are needed to be re-established. We apply Lions’ concentration-compactness principle [14,15] to give a more detailed analysis for the compactness of our problem. Thanks to the new version of symmetric Mountain Pass Lemma in [11], we give the proof of our main result. As far as we know, there are few results on this question, so the research in this paper is meaningful.

Now we first give the definition of weak solutions for problem (1.1).

Definition 1.1. We say thatu ∈W^1,p(_R^N)∩L^∞_loc(_R^N)is a weak solution of (1.1), if Z

R^N

|∇u|^p−2∇u∇ϕ+V(x)|u|^p−2uϕ dx + (_2ω)^p−1

Z

R^N|u|^p(2ω−1)|∇u|^p−2∇u∇ϕdx+ (_2ω)^p−1

Z

R^N|∇u|^p|u|^{p(2ω−1)−2}uϕdx

−a Z

R^Nk(x)|u|^q−2uϕdx−b Z

R^N|u|^2ωp∗−2uϕdx=0 for any ϕ∈C₀^∞(_R^N).

In the sequel we will omit the term weak when referring to solutions that satisfy the conditions of Definition1.1. Our main result of this paper is stated as follows.

Theorem 1.2. Suppose that (_V)and(_K)hold,1≤2ω <q<2ωp.Then

(i) ∀b> 0,∃ a₀ > 0 such that if0 < a < a₀,problem(1.1)has a sequence of solutions {u_n}with I(u_n)<_0, I(u_n)→₀andlimn→_∞u_n=_0.

(ii) ∀a > 0,∃ b₀ > 0such that if 0< b< b₀,problem(1.1) has a sequence of solutions {u_n}with I(un)<0, I(un)→0andlimn→∞un=0.

Remark 1.3. From Theorem1.2it is natural to raise the open problems: What if 2ωp< q< p^∗? This problem would be investigated by the authors in future works.

The outline of this paper is as follows. Reformulation of the problem and some prelim- inaries are given in the forthcoming section. In Section 3, behavior of (PS) sequences are established. The proof of Theorem1.2is given in Section 4.

We denote thatL^p(_R^N)is the usual Lebesgue space with the normkuk^p_p =R

R^N|u|^pdx, 1≤ p < +_∞. kuk^p = R

R^N|∇u|^pdx,kuk^p_V = R

R^N(|∇u|^p+V(x)|u|^p)dx.S = inf_u∈W1,p(_R^N)\{0} kuk^p kuk^p_p∗

is the best Sobolev constant. Various positive constants are denoted byCandC_i.

2 Reformulation of the problem and preliminaries

The purpose of this section is to establish the variational structure of (1.1) and the main difficulty arises from the function space where the energy functional (1.2) is not well defined inW^1,p(_R^N). For example, if 1< p< Nanduis defined by

u(x) =|x|^(p−N)/2ωp forx ∈B₁\ {0}, we then have thatu∈W^1,p(_R^N)_{, but}

Z

R^N|u|^p(2ω−1)|∇u|^pdx= +_∞.

To overcome this difficulty, we employ an argument developed by Liu, Wang and Wang in [17]

or Colin and Jeanjean in [5]. We use the change of variablesv= f⁻¹(u), where f is defined by

f⁰(t) = ¹

[₁+ (_2ω)^p−1|f(t)|^p(2ω−1)]^1p ,

and f(0) = 0 on [0,+_∞) and by f(t) = −f(−t) on (−_∞, 0]. The following result is due to Adachi and Watanabe in [1] which collects some properties of f .

Lemma 2.1. The function f(t)enjoys the following properties:

(1) f is uniquely defined C^∞ function and invertible.

(2) |f⁰(t)| ≤1,|f(t)| ≤(2ω)^2ωp1 |t|^2ω1 for all t∈_R.

(₃) ^f(_t^t) →₁as t→_0.

(4) ^f(t)

t^2ω1

→a>0as t→+_∞.

(5) _2ω¹ f(t)≤t f⁰(t)≤ f(t)for all t≥0.

(6) There exists a positive constant C such that

|f(t)| ≥







C|t|_, |t| ≤_1, C|t|^2ω1 , |t|>1.

After the above change of variables, we can rewrite our energy functional (1.2) in the terms ofv:

J(v) = ¹ p

Z

R^N

|∇v|^p+V(x)|f(v)|^pdx− ^a q

Z

R^Nk(x)|f(v)|^qdx− ^b 2ωp^∗

Z

R^N|f(v)|^2ωp∗dx.

We first give the proof of the following weakly continuous lemma.

Lemma 2.2.

(i) The functionalF(v) =R

R^Nk(x)|f(v)|^qdx is well defined and weakly continuous on W^1,p(_R^N). Moreover, F(v) is continuously differentiable, its derivativeF⁰ : W^1,p(_R^N) → (W^1,p(_R^N))^∗ is given by

hF⁰(v),gi=q Z

R^Nk(x)|f(v)|^q−2f(v)f⁰(v)gdx, ∀g ∈W^1,p(_R^N).

(ii) The functionalG(v) =R

R^N f(v)^2ωp∗dx is well defined. Moreover,G(v)is continuously differ- entiable, its derivativeG⁰ :W^1,p(_R^N)→(W^1,p(_R^N))^∗ is given by

hG⁰(v),gi=2ωp^∗ Z

R^N|f(v)|^2ωp∗−2f(v)f⁰(v)gdx, ∀g ∈W^1,p(_R^N).

Proof. Firstly, by (3) and (4) in Lemma 2.1, it is clear that F(v) and G(v) are well defined on W^1,p(_R^N). Next, we prove that F(_v)_,G(v) ∈ C¹(_R^N). It suffices to show that both F(_v) and G(v) have continuous Gateaux derivatives on W^1,p(_R^N). We only prove that F(v) has continuous Gateaux derivatives on W^1,p(_R^N)since the case of the proof for G(v)is simpler.

Our proof is the same as the proof of Lemma 3.10 in [22] , for the convenience of the readers, we present the process. Let v,g ∈ W^1,p(_R^N). Given 0 < |t| <1, by the mean value theorem, there existsλ∈(0, 1)such that

||f(v+tg)|^q− |f(v)|^q|

|t| =q|f(v+tλg)|^q−1|f⁰(v+tλg)||g|

=q|f(v+tλg)|^q|f⁰(v+tλg)|

|f(v+tλg)||g|

≤C|v+tλg|^2ωq |v+tλg|⁻¹|g|

=C|v+tλg|^q−2ω2ω|g|

≤C(|v|^q−2ω2ω|g|+|g|^2ωq ),

where the conclusions of Lemma 2.1 (2) and (5) are used. By the Hölder inequality and assumption of(K), we have

Z

R^Nk(x)(|v|^q−2ω2ω|g|+|g|^2ωq )dx ≤ kk(x)k_rkgk_p^∗(kvk^q−2ω2ω +kgk^q−2ω2ω).

It follows from the Lebesgue Dominated Convergence Theorem that F(v) is Gateaux differ- entiable and

hF⁰(v),gi=q Z

R^Nk(x)|f(v)|^q−2f(v)f⁰(v)gdx.

Now, we give the proof of continuity of Gateaux derivative. Assume thatvn→vinW^1,p(_R^N), then f²(v_n)→ f²(v)inW^1,p(_R^N). By the continuity of the embeddingW^1,p(_R^N),→L^p∗(_R^N), we get that f²(v_n) → f²(v) in L^p∗(_R^N). Define K(v) = k(x)|f(v)|^q−2f(v)f⁰(v). Then K ∈ (L^p∗(_R^N),C(L^p∗(_R^N))⁰). It follows that K(vn) → K(v) in (L^p∗(_R^N))⁰. Using the Hölder and Sobolev inequalities, we have

hF⁰(v_n)− F⁰(v),gi ≤ kK(v_n)− K(v)k_(p∗)⁰kgk_p^∗ ≤CkK(v_n)− K(v)k_(p∗)⁰kgk. HencekF⁰(vn)− F⁰(v)k →0 andF ∈C¹.

From the above analysis we can get that J(v)is well defined on W^1,p(_R^N) under the as- sumptions of(V)and(K). The standard arguments applied in [20,22] show that J(v)belongs to C¹(W^1,p(_R^N)_,R). As in [5], we note that if v is a nontrivial critical point of J, v then is a nontrivial solution of the problem

−_∆pv+V(x)|f(v)|^p−2f(v)f⁰(v) =ak(x)|f(v)|^q−2f(v)f⁰(v) +b|f(v)|^2ωp∗−2f(v)f⁰(v). (2.1) Therefore, let u = f(v)and since (f⁻¹)⁰(t) = [1+ (2ω)^p−1|f(t)|^p(2ω−1)]^1p, we conclude thatu is a nontrivial solution of the problem (1.1).

Now we can restate Theorem1.2as follows.

Theorem 2.3. Suppose that(V)and(K)are held,ω>1/2, 2ω <q<2ωp. Then

(i) ∀b >0,∃ a_e0 > 0such that if0 < a < a_e0,problem(2.1)has a sequence of solutions{v_n}with J(vn)<0,J(vn)→0andlimn→_∞vn =0.

(ii) ∀a >0,∃be₀ > 0such that if0< b < be₀,problem(2.1)has a sequence of solutions{vn}with J(v_n)<0,J(v_n)→0andlim_n→_∞v_n =0.

3 Properties of ( PS )

sequences

In this section, we perform a careful analysis of the behavior of minimizing sequences with the aid of Lions’ concentration–compactness principle [14,15], which allows us to recover the compactness below some critical threshold.

LetEbe a real Banach space andJ : E→_Rbe a function of classC¹. We say that{vn} ⊂E is a(PS)_csequence if J(v_n)→cand J⁰(v_n)→0. J is said to satisfy the Palais–Smale condition at levelc((PS)_cfor short) if any(PS)_c sequence contains a convergent subsequence.

Lemma 3.1. Assume(_V)and(_K),{v_n} ⊂W^1,p(_R^N)be a(PS)_c sequence for J at level c < 0and 2ω< q<2ωp.Then

(i) there exists C>0such that, for all n∈_N,kvnk_V≤ C;

(ii) ∀b>0,∃ a∗ >0such that if0< a<a∗,then J satisfies(PS)_c; (iii) ∀a>0,∃b∗ >0such that if0<b<b∗,then J satisfies(PS)_c.

Proof. At first, we prove that {v_n}is bounded in W^1,p(_R^N). Let {v_n}be a(PS)_c sequence in W^1,p(_R^N)such that for allφ∈C₀^∞(_R^N), we have that

c+on(kvnk) = J(vn) = ¹ p

Z

R^N

|∇vn|^p+V(x)|f(vn)|^pdx

−^a q

Z

R^Nk(x)|f(v_n)|^qdx− ^b 2ωp^∗

Z

R^N|f(v_n)|^2ωp∗dx,

(3.1)

and

o_n(kv_nk) =hJ⁰(v_n),ϕi=

Z

R^N

|∇v_n|^p−2∇v_n∇ϕ+V(x)|f(v_n)|^p−2f(v_n)f⁰(v_n)ϕ

dx

−a Z

R^Nk(x)|f(vn)|^q−2f(vn)f⁰(vn)ϕdx

−b Z

R^N|f(v_n)|^2ωp∗−2f(v_n)f⁰(v_n)ϕdx.

(3.2)

Chooseϕ= ϕn = [1+ (2ω)^p−1|f(vn)|^p(2ω−1)]^1pf(vn), we haveϕn∈W^1,p(_R^N)and then|ϕn|<

2ω|v_n|. Since

∇ϕ_n =

"

1+ (2ω−1)(2ω)^p−1|f(v_n)|^p(ω−1) 1+ (2ω)^p−1|f(vn)|^p(ω−1)

#

∇v_n≤(2ω)∇v_n, f⁰(v_n)ϕ_n= f(v_n), we getkϕ_nk ≤Ckv_nk. It follows from (3.2) that

o_n(kv_nk) =hJ⁰(v_n)_,ϕ_ni ≤

Z

R^N(_2ω|∇v_n|^p+V(x)|f(v_n)|^p)dx

−a Z

R^Nk(x)|f(vn)|^qdx−b Z

R^N|f(vn)|^2ωp∗dx.

(3.3)

Since

|∇f^2ω(v_n)|^p =|_2ωf^2ω−1(v_n)f⁰(v_n)∇v_n|^p

= (2ω)^p |f^2ω−1(v_n)|^p

1+ (2ω)^p−1|f(vn)|^p(2ω−1)|∇v_n|^p

=

"

2ω (2ω)^p−1|f(v_n)|^p(2ω−1) 1+ (2ω)^p−1|f(vn)|^p(2ω−1)

#

|∇v_n|^p

≤2ω|∇v_n|^p,

(3.4)

we get

0>c+o_n(kv_nk) = J(v_n)− ¹

2ωp^∗hJ⁰(v_n)_,ϕ_ni

≥ ¹ N

Z

R^N|∇v_n|^pdx+ 1

p− ¹ 2ωp^∗

_Z

R^NV(x)|f(v_n)|^pdx− ^a qr

Z

R^Nk(x)|f(v_n)|^qdx

≥ ¹ N

Z

R^N(|∇v_n|^p+V(x)|f(v_n)|^p)dx− ^a

qrkk(x)k_{r Z}

R^N|f^2ω(v_n)|^p∗dx _2ωp^q_∗

≥ ¹ N

Z

R^N(|∇vn|^p+V(x)|f(vn)|^p)dx− ^a qrC₁

Z

R^N|∇f^2ω(vn)|^pdx _2ωp^q

≥ ¹ N

Z

R^N(|∇v_n|^p+V(x)|f(v_n)|^p)dx−C_{2 Z}

R^N|∇v_n|^pdx _2ωp^q

≥ ¹ N

Z

R^N(|∇v_n|^p+V(x)|f(v_n)|^p)dx−C_{2 Z}

R^N(|∇v_n|^p+V(x)|f(v_n)|^p)dx _2ωp^q

,

(3.5)

which implies that fornlarge enough, there existsC>0 such that Z

R^N(|∇v_n|^p+V(x)|f(v_n)|^p)dx ≤C. (3.6) In the following, we need to show{v_n}is bounded inW^1,p(_R^N). From (3.6), we need to prove that R

R^NV(x)|vn|^pdxis bounded. By(V), Z

{x:|vn|>1}V(x)|v_n|^pdx≤ M Z

{x:|vn|>1}

|v_n|^p∗dx≤ MS^−p

∗ p

_Z

{x:|vn|>1}

|∇v_n|^pdx ^p

∗ p

, and using Lemma2.1 (6),

Z

{x:|vn|≤1}V(x)|v_n|^pdx≤ ¹ C²

Z

{x:|vn|≤1}V(x)|f(v_n)|^pdx≤ ¹ C²

Z

R^NV(x)|f(v_n)|^pdx.

These estimates imply that {v_n} is bounded in W^1,p(_R^N). Then {f(v_n)} is also bounded in W^1,p(_R^N). Therefore we can assume that

v_n*v weakly inW^1,p(_R^N)_, vn→v a.e. in R^N,

v_n→v strongly inL_loc^t (_R^N)for allt∈ [1,p^∗). Since f ∈C^∞, we have

f^2ω(v_n)* f^2ω(v) weakly inW^1,p(_R^N), f^2ω(v_n)→ f^2ω(v) a.e. inR^N.

In view of the concentration–compactness principle [14,15], there exist a subsequence, still denoted by {f(vn)}, µ,ν ∈ M(_R^N ∪ {_∞}) which are the positive finite Radon measures on R^N∪ {_∞}, an at most countable setJ, a set of different points {x_j} ⊂_R^N, and real numbers µ_j,ν_jsuch that the following convergence hold in the sense of measures

|∇f^2ω(v_n)|^p*dµ≥ |∇f^2ω(v)|^p+

∑

j∈J

µ_jδ_xj,

|f^2ω(vn)|^p∗ *dν= |f^2ω(v)|^p∗+

∑

j∈J

ν_jδx_j.

From the above two equations and the Sobolev inequalities, it follows easily that µ_j ≥Sν

p p∗

j for allj∈ J. (3.7)

Concentration at infinity of the sequence{u_n}is described by the following quantities:

µ_∞ := lim

R→_∞

lim sup

n→_∞ Z

{x:|x|>R}|∇f^2ω(v_n)|^pdx, ν_∞ := lim

R→_∞

lim sup

n→_∞ Z

{x:|x|>R}|f^2ω(vn)|^p∗dx.

We claim that

J is finite and, for j∈ J, eitherν_j =0 or ν_j ≥(b⁻¹S)^N/2.

In fact, for ε > 0, letting x_j be a singular point of the measures µ_j and ν_j, φ_j(x) be a smooth cut–off function centered at x_j such that 0 ≤ φ_j(x) ≤ 1, φ_j(x) ≡ 0 on |x−x_j| ≥ 2, φ_j(x) ≡ 1 on |x−x_j| ≤ 1, and |∇φ_j(x)| ≤ 2 for all x ∈ _R^N. Letting φ^ε_j(x) = φ_j(^x

ε),ψ_n = [1+ (2ω)^p−1|f(v_n)|^p(2ω−1)]^1pf(v_n), then we get that {ψ_n} is bounded in W^1,p(_R^N). Testing J⁰(vn)withψnφ^ε_j, we obtain limn→_∞hJ⁰(vn),ψnφ^ε_j(x)i=0, that is

− lim

n→_∞ Z

R^N[1+ (2ω)^p−1|f(v_n)|^p(2ω−1)]^1pf(v_n)|∇v_n|^p−2∇v_n∇φ^ε_jdx

= lim

n→_∞

Z

R^N

1+2ω(2ω)^p−1|f(vn)|^p(2ω−1)

1+ (2ω)^p−1|f(v_n)|^p(2ω−1) |∇vn|^pφ^ε_jdx+

Z

R^NV(x)|f(vn)|^pφ^ε_jdx

−a Z

R^Nk(x)|f(v_n)|^qφ^ε_jdx−b Z

R^N|f(v_n)|^2ωp∗φ^ε_jdx

.

(3.8)

In the following we estimate each term in (3.8). By Lemma2.1(5) and the expression of f⁰, we have

|f(v_n)|

f⁰(v_n) ≤2ω|v_n| ⇒[1+ (2ω)^p−1|f(v_n)|^p(2ω−1)]^1p|f(v_n)| ≤2ω|v_n|. Thus

0≤ lim

ε→0 lim

n→_∞

Z

R^N[1+ (2ω)^p−1|f(vn)|^p(2ω−1)]^1p f(vn)|∇vn|^p−2∇vn∇φ^ε_jdx

≤ lim

ε→0 lim

n→_∞ Z

R^N

2ωvn|∇vn|^p−2∇vn∇φ^ε_j dx

≤ lim

ε→0 lim

n→_∞2ω^Z

R^N|∇vn|^pdx^p

−1 p ^Z

R^N|vn∇φ^ε_j|^pdx¹_p

≤ lim

ε→0 lim

n→_∞C^Z

R^N|v_n∇φ^ε_j|^pdx¹_p

≤ Clim

ε→0

^Z

B(x_j,2ε)

|v_n|^p·p

∗

p dx_p¹∗^Z

B(x_j,2ε)

|∇φ^ε_j|^p·Npdx_N¹

=0.

(3.9)

Also we have

nlim→_∞ Z

R^N|∇f^2ω(v_n)|^pφ^ε_jdx=

Z

R^Nφ^ε_jdx≥

Z

R^N|∇f^2ω(v)|^pφ^ε_jdx+µ_j, (3.10) and

nlim→_∞ Z

R^N|f(v_n)|^2ωp∗φ^ε_jdx =

Z

R^Nφ^ε_jdν≥

Z

R^N|f(v)|^2ωp∗φ^ε_jdx+ν_j. (3.11) By the weak continuity ofF(v), we get

lim

ε→0 lim

n→_∞ Z

R^Nk(x)|f(v_n)|^qφ^ε_jdx=0. (3.12) From (3.9)–(3.12), by the weak continuity ofF, we have

0= lim

ε→0 lim

n→_∞

_Z

R^N

1+2ω(2ω)^p−1|f(vn)|^p(2ω−1)

1+ (2ω)^p−1|f(vn)|^p(2ω−1) |∇v_n|^pφ^ε_jdx+

Z

R^NV(x)|f(v_n)|^pφ^ε_jdx

−a Z

R^Nk(x)|f(vn)|^qφ^ε_jdx−b Z

R^N|f(vn)|^2ωp∗φ^ε_jdx

≥ lim

ε→0 lim

n→_∞

Z

R^N|∇f^2ω(vn)|^pφ^ε_jdx−a Z

R^Nk(x)|f(vn)|^qφ^ε_jdx−b Z

R^N|f(vn)|^2ωp∗φ^ε_jdx

=µ_j−bν_j.

(3.13)

Combining with (3.7), we obtain

either (i)ν_j =0 or (ii)ν_j ≥(b⁻¹S)^Np, which implies thatJ is finite. The claim is thereby proved.

To analyze the concentration at ∞, we follow closely the argument used in [21]. By choosing a suitable cut-off function ϕ ∈ C₀^∞(_R^N,[0, 1]) such that ϕ(x) ≡ 0 on |x| ≤ 1 and ϕ(x) ≡ 1 on |x| ≥ 2. Setting ϕR(x) = ϕ(^x_R), then {ϕRψn} is bounded in W^1,p(_R^N), and lim_n→_∞hJ⁰(v_n),ϕ_Rψ_ni=0, that is

− lim

n→_∞ Z

R^N[1+ (2ω)^p−1|f(v_n)|^p(2ω−1)]^1pf(v_n)|∇v_n|^p−2∇v_n∇ϕ_Rdx

= lim

n→_∞

_Z

R^N

1+2ω(2ω)^p−1|f(v_n)|^p(2ω−1)

1+ (2ω)^p−1|f(vn)|^p(2ω−1) |∇v_n|^pϕ_Rdx+

Z

R^NV(x)|f(v_n)|^pϕ_Rdx

−a Z

R^Nk(x)|f(vn)|^qϕ_Rdx−b Z

R^N|f(vn)|^2ωp∗ϕ_Rdx

.

(3.14)

Similar to the process of (3.9), we can get

Rlim→_∞ lim

n→_∞ Z

R^N[1+ (2ω)^p−1|f(v_n)|^p(2ω−1)]^1pf(v_n)|∇v_n|^p−2∇v_n∇ϕ_Rdx=0. (3.15) Using the weak continuity ofF, we have

Rlim→_∞ lim

n→_∞ Z

R^Nk(x)|f(vn)|^qϕ_Rdx =0.

Therefore,

0= _lim

R→_∞ lim

n→_∞

Z

R^N

1+2ω(2ω)^p−1|f(v_n)|^p(2ω−1)

1+ (2ω)^p−1|f(v_n)|^p(2ω−1) |∇v_n|^pϕ_Rdx+

Z

R^NV(x)|f(v_n)|^pϕ_Rdx

−a Z

R^Nk(x)|f(v_n)|^qϕ_Rdx−b Z

R^N|f(v_n)|^2ωp∗ϕ_Rdx

≥ lim

R→_∞ lim

n→_∞

_Z

R^N|∇f^2ωv_n|^pϕ_Rdx−a Z

R^Nk(x)|f(v_n)|^qϕ_Rdx−b Z

R^N|f(v_n)|^2ωp∗ϕ_Rdx

=µ_∞−bν_∞. (3.16)

By

µ_∞ ≥Sν

p p∗

∞ , (3.17)

we get

either (iii)ν_∞ =0 or (iv)ν_∞≥ (b⁻¹S)^Np.

Next, we claim that (ii) and (iv) cannot occur if a and b are chosen properly. In fact, by (3.4) and(V), we have

Z

R^N(|∇v|^p+V(x)|f(v)|^p)dx ≥

Z

R^N|∇v|^pdx ≥ ¹ 2ω

Z

R^N|∇f^2ω(v)|^pdx.

Then if (iv) holds, from the weak lower semicontinuity of the norm and the weak continuity ofF, we have,

0>c= lim

n→_∞

J(v_n)− ¹

2ωp^∗hJ⁰(v_n),ϕ_ni

≥ lim

n→_∞

1 p− ¹

p^∗ Z

R^N(|∇vn|^p+V(x)|f(vn)|^p)dx− ^a

qrkk(x)k_rkf^2ω(vn)k

q 2ω

p^∗

≥ ¹ N

Z

R^N(|∇v|^p+V(x)|f(v)|^p)dx− ^a

qrkk(x)k_rkf^2ω(v)k

q

p2ω^∗

≥ ¹ N · ¹

2ω Z

R^N|∇f^2ω(v)|^pdx− ^a

qrkk(x)k_rkf^2ω(v)k

q

p2ω^∗

≥ ¹

2ωNSkf^2ω(v)k^p_p∗− ^a

qrkk(x)k_rkf^2ω(v)k_p^2ωq∗. This inequality implies that

kf^2ω(v)k_p^∗ ≤Ca^2ωp2ω−q. Therefore from (3.17) and (iv),

0>c= lim

n→_∞

J(v_n)− ¹

2ωp^∗hJ⁰(v_n),ϕ_ni

≥ lim

n→_∞

1 p− ¹

p^∗ _Z

R^N(|∇vn|^p+V(x)|f(vn)|^p)dx− ^a

qrkk(x)k_rkf^2ω(vn)k

q 2ω

p^∗

≥ lim

R→_∞ lim

n→_∞

1 N

Z

R^N(|∇v_n|^p+V(x)|f(v_n)|^p)φ_Rdx− ^a

qrkk(x)k_rkf^2ω(v_n)k_p^2ωq∗

≥ ¹

Nµ_∞S^Np −Ca^2ωp2ω−q

≥ ¹

Nb^p−pNS^Np −Ca^2ωp2ω−q.

However, if a> 0 is given, we can choose smallb∗ so that for every 0 <b< b∗, the last term on the right-hand side above is greater than zero, which is a contradiction. Similarly, ifb>0 is given, we can take smalla∗ so that for every 0<a< a∗, the last term on the right-hand side above is greater than zero. Similarly, we can prove that (ii) cannot occur for eachj. Hence

kf^2ω(v_n)k_p^∗ → kf^2ω(v)k_p^∗ asn→_∞,

and _Z

R^Nk(x)(|f(v_n)|^q− |f(v)|^q)dx≤ kk(x)k_rk|f(v_n)|^q− |f(v)|^qk2ωp∗ q

. Thus, from the weak lower semicontinuity of the norm andF ∈C^∞ we have

o(kv_nk) =hJ⁰(v_n)_,ϕ_ni

=

Z

R^N(|∇v_n|^p+V(x)|f(v_n)|^p)dx+

Z

R^N

"

(_2ω−₁)(_2ω)^p−1|f(v_n)|^p(2ω−1) 1+ (2ω)^p−1|f(v_n)|^p(2ω−1)

#

|∇v_n|^pdx

−a Z

R^Nk(x)|f(v_n)|^qdx−b Z

R^N|f(v_n)|^2ωp∗dx

=

Z

R^N(|∇vn− ∇v|^p+|∇v|^p+V(x)|f(v)|^p)dx +

Z

R^N

"

(2ω−1)(2ω)^p−1|f(v)|^p(2ω−1) 1+ (2ω)^p−1|f(v)|^p(2ω−1)

#

|∇v|^pdx

−a Z

R^Nk(x)|f(v)|^qdx−b Z

R^N|f(v)|^2ωp∗dx+o(kv_nk)

=

Z

R^N|∇v_n− ∇v|^pdx+o(kv_nk)

since J⁰(v) =0. Thus we prove that{v_n}strongly converges tovinW^1,p(_R^N)_.

4 Proofs of the main results

In this section, we use the minimax procedure (see [20]) to prove the existence of infinitely many solutions. Let X be a Banach space andΣ be the class of subsets of X\ {0}which are closed and symmetric with respect to the origin. For A∈_Σ, we define the genusγ(A)by

γ(A) =min{n∈ _N:∃φ∈C(A,Rⁿ\ {0}), φ(z) =−φ(−z)}.

If there is no mapping as above for anyn∈_{N, then}γ(A) = +_{∞. LetΣn}denote the family of closed symmetric subsets AofXsuch that 06∈ Aandγ(A)≥n. We list some properties of the genus (see [11,20]).

Proposition 4.1. Let A and B be closed symmetric subsets of X which do not contain the origin. Then the following hold:

(i) If there exists an odd continuous mapping from A to B,thenγ(A)≤γ(B); (ii) If there is an odd homeomorphism from A to B,thenγ(A) =γ(B);

(iii) Ifγ(B)<_∞,thenγ(A\B)≥γ(A)−γ(B);

(iv) n-dimensional sphere S_nhas a genus of n+₁by the Borsuk–Ulam Theorem;