Infinitely many solutions for a quasilinear Schrödinger equation with Hardy potentials

Infinitely many solutions for a quasilinear Schrödinger equation with Hardy potentials

Tingting Shang and Ruixi Liang

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P.R.China Received 28 April 2020, appeared 26 July 2020

Communicated by Patrizia Pucci

Abstract. In this article, we study the following quasilinear Schrödinger equation

−∆u−µ u

|x|²+V(x)u−(_∆(u²))u= f(x,u)_, x∈R^N,

where V(x)is a given positive potential and the nonlinearity f(x,u) is allowed to be sign-changing. Under some suitable assumptions, we obtain the existence of infinitely many nontrivial solutions by a change of variable and Symmetric Mountain Pass The- orem.

Keywords: quasilinear Schrödinger equation, Hardy potential, Mountain Pass Theo- rem.

2020 Mathematics Subject Classification: 35J20, 35J60.

1 Introduction and main results

In this paper, we consider the following equation

−_∆u−µ u

|x|² +V(x)u−(_∆(u²))u= f(x,u), x∈_R^N, (1.1) where N ≥ 3, 0 ≤ µ < µ¯ := ^(N−₄²⁾², V(x) ∈ C(_R^N,R) is a given potential and f ∈ C(_R^N×_R,R).

For problem (1.1), ifµ=0, f(x,u) = f(u), then (1.1) becomes

−_∆u+V(x)u−(_∆(u²))u= f(u), x∈ _R^N. (1.2) Recently, the existence of solutions for (1.2) has drawn much attention, see for example [5,7, 19,21,22,25]. Particularly, it was established the existence of both one-sign and nodal ground states of soliton type solutions in [21] by Nehari method. Furthermore, using a constrained minimization argument, the existence of a positive ground state solution has been proved in [25]. Later, by using a change of variables, [19] and [7] studied the existence of solutions in

BCorresponding author. Email: lrxcsu@163.com

different working spaces with different classes of nonlinearities. For more results we can refer to [18,20,23,33,34].

Moreover, if we takeµ≡0 in (1.1), we have

−_∆u+V(x)u−(_∆(u²))u= f(x,u)_, x ∈_R^N_{. (1.3)} In [39], Zhang and Tang proved there are infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential by Mountain Pass Theorem. When f(x,u) = ^|u|2

∗(s)−2u

|x|^s , where 0 ≤ s < 2 and 2^∗(s) = ²⁽_N^N₋⁻₂^s) is the critical Sobolev–Hardy exponents, the problem (1.3) was studied in [10,12]. If f(x,u) =λ|u|^q−2u+^|u|_|^p−2u

x|^s , the authors in [40] have proved the existence of solutions by using a change of variable.

Recently, great attention has been attracted to the study of the following problem

−_∆u−µ u

|x|² +V(x)u= f(x,u), x∈_R^N. (1.4) This class of quasilinear equations are often referred as modified nonlinear Schrödinger equa- tions, whose solutions are related to the existence of standing wave solutions. For example, by use of variational method, Kang and Deng in [13] proved the existence of solutions for V(x) =0 and f(x,u) = ^|u|2

∗(s)−2u

|x|^s +K(x)|u|^r−2u. Using the similar method, Li in [14] proved the existence of nontrivial solutions forV(x) = 0 and f(x,u) = ^|u|2

∗(s)−2u

|x|^s +K(x)|u|^r−2u+λu.

In [4], Cao and Zhou studied the problem (1.4) withV(x) ≡ 1 and general subcritical non- linearity f(x,u), they obtained the existence and multiplicity of positive solutions in some different conditions, their method relies upon the proof of Tarantello in [30]. Under certain conditions, using Ekeland’s variational principle, Chen and Peng in [6] obtained the existence of a positive solution with V(x) ≡ 1 and nonlinearity λ(f(x,u) +h(x)). For more results about (1.4), we can refer to [9,11,29] and the references therein.

As regards other relevant papers, we mention here [8,15–17,27,28,31,35,38]. Motivated by facts mentioned above, in this paper, we study the existence of infinitely many solutions for problem (1.1) by Mountain Pass Theorem. Before giving the main result of this paper, we give the assumptions of the potentialV(x)and the nonlinear term f(x,u)as follows, firstly

(V₁) V∈C(_R^N,R)and inf

x∈_R^NV(x) =V₀ >0;

(V₂) for anyL>0, there exists a constantϑ>0 such that

|ylim|→_∞meas{x∈_R^N :|x−y| ≤ϑ,V(x)≤L}=0;

(F₀) f ∈C(_R^N×_R,R)and there exist constantsc₁,c₂ >0 and 4< p<22^∗ such that

|f(x,u)| ≤c₁|u|+c₂|u|^p−1, ∀(x,u)∈_R^N×_R;

(F₁) lim

|u|→_∞ F(x,u)

u⁴ = _∞ uniformly in x, and there exists a₀ ≥ 0 such that F(x,u) ≥ 0 for all (x,u)∈_R^N×_Rand|u| ≥a₀, whereF(x,u) =Ru

0 f(x,s)ds;

(F₂) F^˜(x,u):= ¹₄f(x,u)u−F(x,u)≥0 and there existc₀>0 andσ ∈ max{1,_N^2N₊₂}, 2 such that

|F(x,u)|^σ≤c₀|u|^2σF^˜(x,u) for all(x,u)∈_R^N×_Rwith ularge enough;

(F₃) f(x,u) =−f(x,−u)for all (x,u)∈_R^N×_R.

Now, we are ready to state the main result of this paper.

Theorem 1.1. Assume that(V₁)–(V₂),(F₀)–(F₃)are satisfied, then problem(1.1)has infinitely many nontrivial solutions{u_n}such thatku_nk →_∞and I(u_n)→_∞(I will be defined later).

Remark 1.2(see [30]). It follows from(F₁)and(F₂)that F˜(x,u)≥ ¹

c₀

|F(x,u)|

|u|² σ

→_∞, (1.5)

uniformly inxas |u| →_∞.

This paper is organized as follows. In Section 2, we will introduce the variational setting for the problem and some preliminary results. In Section 3, we give the proof of main result.

Notations. In what follows we will adopt the following notations

• C,C_i,i=1, 2, 3, . . . denote possibly different positive constants which may change from line to line;

• For 1≤ p<_∞, L^p(_R^N)denotes the usual Lebesgue spaces with norms kuk_p =

_Z

R^N|u|^pdx 1/p

, 1≤ p<_∞;

• H¹(_R^N)denotes the Sobolev spaces modeled in L²(_R^N)with norm kuk_H1 =

_Z

R^N|∇u|²+|u|²dx 1/2

.

• B_R denotes the open ball centered at the origin and radiusR>0.

2 Variational setting and preliminary results

Before establishing the variational setting for problem (1.1), we give our working space firstly.

Under the assumption(V₁)we define E:=

u∈ H¹(_R^N): Z

R^NV(x)u²dx<_∞

, then Eis a Hilbert space equipped with the inner product and norm

(u,v) =

Z

R^N

∇u∇v−µ uv

|x|² +V(x)uv

dx, kuk= (u,u)^1/2. In view of (V₁)and foru∈ E, the following norm

kuk_E = _Z

R^N(|∇u|²+V(x)u²)dx ¹₂

is equivalent to the classic one in H¹(_R^N).

Now, let us recall the Hardy inequality, which is the main tool and allows us to deal with Hardy-type potentials.

Lemma 2.1(see [1]). Assume that1< p< N and u∈W^1,p(_R^N), then Z

R^N

|u|^p

|x|^pdx≤

p (N−p)

pZ

R^N|∇u|^pdx.

Thus, by Lemma 2.1,kukis well defined. In fact Z

R^N

|∇u|²−µ u²

|x|²

dx

≥

Z

R^N

|∇u|²−µ 4

(N−2)²|∇u|²

dx

=

1−µ 4 (_N−2)²

_Z

R^N|∇u|²dx

>

1− (N−2)² 4

4 (N−2)²

_Z

R^N|∇u|²dx

=0.

(2.1)

Lemma 2.2. Assume that0≤µ<µ¯ = ^(N−₄²⁾², then there exist C₁,C₂>0such that C₁kuk²_E ≤ kuk² ≤C₂kuk²_E,

for any u∈ H¹(_R^N). Proof. Forµ≥0, we have

kuk²=

Z

R^N

|∇u|²−µ u²

|x|² +V(x)u²

dx

≤

Z

R^N(|∇u|²+V(x)u²)dx

= kuk²_E.

(2.2)

On the other hand, since 0≤µ<µ¯ = ^(N−₄²⁾², we can get 1≥₁− ^4µ

(N−2)² >_0.

Then, we have

kuk²=

Z

R^N

|∇u|²−µ u²

|x|² +V(x)u²

dx

≥

Z

R^N

|∇u|²− ^4µ

(N−2)²|∇u|²+V(x)u²

dx

≥

1− ^4µ (N−2)²

Z

R^N(|∇u|²+V(x)u²)dx

=

1− ^4µ (N−2)²

kuk²_E.

(2.3)

It follows from (2.2) and (2.3) that

C₁kuk²_E ≤ kuk² ≤C₂kuk²_E.

As we all known, under the assumption (V₁), the embedding E ,→ L^r(_R^N)is continuous forr∈ [2, 2^∗]andE,→L^r_loc(_R^N)is compact for[2, 2^∗)i.e. there is a constantdr >0 such that

kuk_s≤d_rkuk_E, ∀u∈E, r ∈[2, 2^∗]. From this, by Lemma 2.2, there isC₃>0 such that

kuk_r≤drkuk_E ≤C₃kuk, ∀u∈E, r ∈[2, 2^∗].

Furthermore, under the assumptions (V₁) and (V₂), we have the following compactness lemma due to [3] (see also [2,41]).

Lemma 2.3. Assume that(V₁)and(V₂)hold, the embedding E ,→L^r(_R^N)is compact for2≤r<2^∗. In order to solve problem (1.1), we define the energy functional I : E→_{Rgiven by}

I(u) = ¹ 2

Z

R^N(1+2|u|²)|∇u|²dx− ¹ 2

Z

R^N

µ

|x|^2u

2dx+¹ 2

Z

R^NV(x)u²dx−

Z

R^NF(x,u)dx.

It is well known that I is not well defined in E. To overcome this difficulty, we make the change of variables byv=h⁻¹(u), wherehis defined by

h⁰(t) = _p ¹

1+2|h(t)|^{2 on} [0,∞), and

h(−t) =−h(t) on(−_{∞, 0}].

Therefore, after the change of variables, fromI(u)we obtain the following functional J(v):= I(h(v))

=¹ 2

Z

R^N|∇v|²dx− ¹ 2

Z

R^N

µ

|x|^2h

2(v)dx+ ¹ 2

Z

R^NV(x)h²(v)dx−

Z

R^NF(x,h(v))dx. (2.4) It is easy to check that J is well defined onE. Under our hypotheses, J ∈C¹(E,R)and

hJ⁰(v),φi=

Z

R^N∇v∇φdx−

Z

R^N

µ

|x|^2h(v)h⁰(v)φdx +

Z

R^NV(x)h(v)h⁰(v)φdx−

Z

R^N f(x,h(v))h⁰(v)φdx.

(2.5)

for all φ∈ E.

Moreover, the critical points of J are the weak solutions of the following equation

−_∆v= _p ¹ 1+2|h(v)|²

f(x,h(v))−V(x)h(v) + ^µ

|x|^2h(v)

inR^N. (2.6) We also observe that ifvis a critical point of the functional J, thenu= h(v)is a critical point of the functional I, i.e.u= h(v)is a solution of (1.1).

Now, let us recall some properties of the change of variablesh:R→_R.

Lemma 2.4. (see [24])The function h(t)and its derivative satisfy the following properties (h₁) h is uniquely defined, C^∞ and invertible;

(h₂) |h⁰(t)| ≤1for all t ∈_R;

(h3) |h(t)| ≤ |t|for all t∈_R;

(h₄) ^h(_t^t) →1as t→0;

(h₅) ^h√(t)

t →2¹⁴ as t→_∞;

(h₆) ^h(₂^t) ≤th⁰(t)≤ h(t)for all t>0;

(h₇) ^h2₂^(t) ≤th(t)h⁰(t)≤h²(t)for all t∈_R;

(h₈) |h(t)| ≤2¹⁴|t|¹² for all t∈_R;

(h₉) there exists a positive constant C such that

|h(t)| ≥

(C|t|, |t| ≤1 C|t|¹², |t| ≥1;

(h₁₀) for eachα>0, there exists a positive constant C(α)such that

|h(αt)|² ≤C(α)|h(t)|²; (h₁₁) |h(t)h⁰(t)| ≤ ^√1

2.

For convenience of our proof, we give the following basic and important definition.

Definition 2.5 (see [36]). Assume that J ∈ C¹(E,R), sequence {un} ⊂ E is called (C)_c se- quence if

J(v_n)→c and (1+kv_nk)J⁰(v_n)→0.

If any (C)_c sequence has a convergent subsequence, we say J satisfies Cerami condition at levelc.

Lemma 2.6. Assume that(V₁),(V₂),(F₀)–(F₂)hold, then any(C)_c-sequence of J is bounded in E for each c∈ _R.

Proof. Let{v_n} ⊂Ebe a(C)_c-sequence of J, we have

J(v_n)→c, (1+kv_nk)J⁰(v_n)→0 asn→_∞. (2.7) Then, there is a constantC₄ >0 such that

J(v_n)− ¹

4hJ⁰(v_n),v_ni ≤C₄. (2.8) Let

kv_nk²_h :=

Z

R^N

|∇v_n|²− ^µ

|x|^2h

2(v_n) +V(x)h²(v_n)

dx.

First, we prove that there existsC₅ >0 such that

kv_nk²_h ≤C₅. (2.9)

On the contrary, we suppose that

kv_nk²_h→_∞.

Taking ˜h(v_n) = _k^h(vn)

vnk_h, thenkh^˜(v_n)k ≤1. Passing to a subsequence, we assume that h˜(v_n)*ν inE,

h˜(v_n)→ν inL^r(_R^N), 2≤r<2^∗, and

h˜(vn)→ν a.e. on R^N. From (2.4) and (2.7), we have

|nlim|→_∞ Z

R^N

|F(x,h(vn))|

kvnk²_h ^dx= ¹

2. (2.10)

On the other hand, set ξ_n = ^h(vn)

h⁰(vn), then there exists C₆ > 0 such that kξ_nk ≤ C₆kv_nk_{. Since} {v_n}is a(C)_c sequence of J, by (2.8) we have

C₆≥ J(vn)− ¹

4hJ⁰(vn),ξni

= ¹ 4

Z

R^N

|∇h(vn)|²− ^µ

|x|^2h

2(vn) +V(x)h²(vn)

dx

+

Z

R^N

1

4f(x,h(vn))h(vn)−F(x,h(vn))

dx

= ¹

4kh(v_n)k²+

Z

R^N

F˜(x,h(v_n))dx

≥

Z

R^N

F˜(x,h(vn))dx.

(2.11)

Takel(a) =inf{F^˜(x,h(v_n))| x ∈ _R^N, |h(v_n)| ≥a}, for a ≥ 0. By (1.5), we have l(a)→ _∞as a→_{∞. For 0}≤ b₁ <b2, let

B_n(b₁,b₂) ={x∈_R^N :b₁≤ |h(v_n(x))|<b₂}. Combining with (2.11) that

C₆≥

Z

Bn(0,a)

F˜(x,h(v_n))dx+

Z

Bn(_a,∞)

F˜(x,h(v_n))dx

≥

Z

Bn(0,a)

F˜(x,h(v_n))dx+l(a)meas{B_n(a,∞)},

from this we get meas{B_n(a,∞)} → _{0 as} a → _∞ uniformly in n. Hence, for r ∈ [_{2, 2}^∗) _and

(h₁₁), there existC,C₇>0 such that Z

Bn(_a,∞)

h˜^r(v_n)dx≤ _Z

Bn(_a,∞)

h˜^22∗(v_n)dx ₂₂^r∗

meas{B_n(a,∞)}^{2222∗ −∗r}

= ¹

kv_nk^r_{h Z}

Bn(a,∞)h^22∗(v_n)dx ₂₂^r∗

meas{B_n(a,∞)}^{2222∗ −∗r}

≤ ^C kv_nk^r_h

_Z

Bn(a,∞)

|∇h²(v_n)|²dx ^r₄

meas{B_n(a,∞)}^{2222∗ −∗r}

≤ ^C6 kvnk^r_h

_Z

Bn(a,∞)

|∇v_n|²dx ^r₄

meas{B_n(a,∞)}^{2222∗ −∗r}

≤C₇kv_nk⁻_h^r/2 meas{B_n(a,∞)}^{2222∗ −∗r} →0,

(2.12)

asa→_∞uniformly inn.

Ifν = 0, then ˜h(v_n)→0 in L^r(_R^N), 2≤ r <2^∗. For any 0< ε < ¹₈, there exista₁, L large enough, such that

Z

Bn(0,a1)

|F(_x,h(_vn))|

|h(vn)|² |h^˜(vn)|²dx≤

Z

Bn(0,a1)

c1|h(_vn)|²+_c2|h(_vn)|^p

|h(vn)|² |h^˜(vn)|²dx

≤(c₁+c₂a₁^p−2)

Z

Bn(0,r1)

|h^˜(v_n)|²dx

≤(c₁+c₂a₁^p−2)

Z

R^N|h^˜(v_n)|²dx

<ε,

(2.13)

forn > L. Setτ⁰ = ^σ

σ−1, since σ ∈ max{1,_N^2N₊₂}, 2

, then 2τ⁰ ∈ (2, 22^∗). Thus, by (F₂)and (2.12) we have

Z

Bn(a₁,∞)

|F(x,h(v_n))|

|h(v_n)|² |h^˜(v_n)|²dx

≤ _Z

Bn(a1,∞)

|F(x,h(v_n))|

|h(v_n)|² σ

dx ¹_σZ

Bn(a1,∞)

|h^˜(v_n)|^2τ0dx ¹

τ0

≤c₀^1σ _Z

Bn(a₁,∞)

F˜(x,h(v_n))dx ¹_σZ

Bn(a₁,∞)

|h^˜(v_n)|^2τ0dx ¹

τ0

≤C8

Z

Bn(a1,∞)

|h^˜(vn)|^2τ0dx ¹

τ0

<ε.

(2.14)

From (2.13) and (2.14), we can get Z

R^N

|F(x,h(v_n))|

kvnk²_h ^dx =

Z

Bn(0,a1)

|F(x,h(v_n))|

|h(vn)|² |h^˜(_vn)|²dx+

Z

Bn(a1,∞)

|F(x,h(v_n))|

|h(vn)|² |h^˜(_vn)|²dx

<2ε< ¹ 4, forn> L, which contradicts (2.10).

Ifν6=0, then meas{B}>0, whereB={x∈_R^N :ν6=0}. Forx∈ B, we have|h(v_n)| →_∞ as n → _{∞. Hence} B⊂ Bn(a0,∞) forn ∈ _Nlarge enough, where a0 is given in (F₁). By (F₁), we have

F(x,h(v_n))

|h(v_n)|⁴ →_∞ asn→_∞.

Using Fatou’s Lemma, then Z

B

F(x,h(vn))

|h(v_n)|^{4 dx}→_∞ asn→_∞. (2.15)

We see from (2.7) and (2.15) 0= lim

n→_∞

c+o(1)

kvnk²_h = lim

n→_∞

J(vn) kvnk²_h

= lim

n→_∞

1 kv_nk²_h

1 2

Z

R^N |∇v_n|²− ^µ

|x|^2h

2(v_n) +V(x)h²(v_n)dx−

Z

R^NF(x,h(v_n))dx

= _lim

n→_∞

1 2 −

Z

Bn(0,a₀)

F(x,h(v_n))

|h(vn)|² |h^˜(v_n)|²dx−

Z

Bn(a₀,∞)

F(x,h(v_n))

|h(vn)|² |h^˜(v_n)|²dx

≤ ¹

2+lim sup

n→_∞

(c₁+c₂a₀^p−2)

Z

R^N|h^˜(v_n)|²dx−

Z

Bn(a₀,∞)

F(x,h(v_n))

|h(vn)|² |h^˜(v_n)|²dx

≤C9−lim inf

n→_∞ Z

B

F(x,h(vn))

|h(v_n)|⁴ |h(vn)h^˜(vn)|²dx

=−_∞,

which is a contradiction. Hence, (2.9) holds.

In order to prove that{v_n}is bounded, we only need to show that there isC₁₀ > 0 such that

kv_nk²_h≥C₁₀kv_nk². (2.16) Arguing indirectly, for a subsequence, we assume ^k_k^v_v^nk2h

nk² → 0, where vn 6= 0 (if not, the result is obvious). Takeξ_n,1 = _k^vn

vnk,η_n,1= ^h_k^2(vn)

vnk², then Z

R^N

|∇ξ_n,1|²− ^µ

|x|^2ηn,1(x) +V(x)η_n,1(x)

dx→0. (2.17)

It follows from(h3)that Z

R^N

|∇ξ_n,1|²− ^µ

|x|^2ηn,1(x) +V(x)η_n,1(x)

dx

=

Z

R^N

|∇ξ_n,1|²− ^µ

|x|² h²(v_n)

kvnk² +V(x)η_n,1(x)

dx

≥

Z

R^N

|∇ξ_n,1|²− ^µ

|x|² v²_n

kv_nk² +V(x)η_n,1(x)

dx

=

Z

R^N

|∇ξ_n,1|²− ^µ

|x|^2ξ

2n,1+V(x)η_n,1(x)

dx

≥0.

(2.18)

Combining (2.1), (2.17) and (2.18), we Z

R^N

|∇ξ_n,1|²− ^µ

|x|^2ξ

2n,1+V(x)η_n,1(x)

dx→0.

Hence Z

R^N

|∇ξ_n,1|²− ^µ

|x|^2ξ

2n,1

dx →0, Z

R^NV(x)η_n,1(x)dx→0 and Z

R^NV(x)ξ²_n,1dx →1.

Similar to the idea of [37], let Bn = {x ∈ _R^N : |vn(x)| ≥C₁₁}, where C₁₁ >0 is independent of n. We suppose that forε > 0, meas{B_n} < ε. If not, there existsε⁰ > 0 and {v_ni} ⊂ {v_n} such that

meas{x∈_R^N :|v_ni(x)| ≥i} ≥ε⁰,

wherei>0 is a integer. Set B_ni = {x∈_R^N :|v_ni(x)| ≥i}. From (2.1),(h₃)and(h₉)we have kvn_ik²_h=

Z

R^N

|∇vn_i|²− ^µ

|x|^2h

2(vn_i) +V(x)h²(vn_i)

dx

≥

Z

R^N

|∇vn_i|²− ^µ

|x|^2v

2ni+V(x)h²(vn_i)

dx

>

Z

R^NV(x)h²(vn_i)dx

> Ciε⁰ →_∞.

asi → ∞, which is a contradiction. For constants C₁₂,C₁₃ > 0, it follows |vn(x)| ≤ C₁₂, (h₉) and(h₁₀)that

C

C₁₂² v²_n≤h² 1

C₁₂v_n

≤ C₁₃h²(v_n). Hence

Z

R^N\Bn

V(x)ξ²_n,1dx≤C₁₄ Z

R^N\Bn

V(x)^h

2(vn) kv_nk ^dx

≤C₁₄ Z

R^NV(x)η_n,1(x)dx→0,

(2.19)

whereC₁₄>0 is a constant. For another, by absolute continuity of integral, there existsε >0 such that

Z

B⁰V(x)ξ²_n,1dx≤ ¹

2, (2.20)

whereB⁰ ⊂_R^N and meas{B⁰}< ε. By (2.19) and (2.20), we have Z

R^NV(x)ξ²_n,1dx=

Z

R^N\Bn

V(x)ξ²_n,1dx+

Z

Bn

V(x)ξ²_n,1dx≤ ¹

2+o(1).

We can get a contradiction. Hence (2.16) holds. Combining (2.9) with (2.16), we complete the proof of this lemma.

Lemma 2.7. Assume that(V₁),(V₂),(F₀)–(F₂)hold, then J satisfies(C)_c-condition.

Proof. Lemma 2.6 implies that{v_n}is bounded in E. For a subsequence, we can assume that vn * v in E. From Lemma 2.3, vn → v in L^r(_R^N)for all 2 ≤ r < 2^∗ and vn → v a.e. onR^N. First, we claim that there existsC₁₅ >0 such that

Z

R^N

|∇(v_n−v)|²+

V(x)− ^µ

|x|²

(h(v_n)h⁰(v_n)−h(v)h⁰(v))

(v_n−v)dx≥C₁₅kv_n−vk²_E (2.21) Indeed, we may assumev_n6=v(otherwise the conclusion is trivial). Set

ξ_n,2= ^vn−v

kv_n−vk ^{and ηn,2}= ^h(v_n)h⁰(v_n)−h(v)h⁰(v)

v_n−v ,

we argue by contradiction and assume that Z

R^N

|∇ξ_n,2|²− ^µ

|x|^2ηn,2(x)ξ²_n,2+V(x)η_n,2(x)ξ²_n,2

dx→0. (2.22)

Since

d

dt(h(t)h⁰(t)) =h(t)h⁰⁰(t) + (h⁰(t))²= ¹

(1+2h²(t))² >0, h(t)h⁰(t)is strictly increasing and for eachC₁₆ >0, there isδ₁ >0 such that

d

dt(h(t)h⁰(t))≥δ₁,

at |t| ≤ C₁₆. From this, we see that η_n,2(x) is positive. On the other hand, forv_n > v, there exists θ∈ (_v,vn)_{such that}

η_n,2= ^h(v_n)h⁰(v_n)−h(v)h⁰(v)

vn−v = ^d

dt(h(θ)h⁰(θ)) = ¹

(1+2h²(θ))² ≤1.

Similarly, we can prove the casev_n <v.

Hence,

η_n,2(x)≤1 for all v_n 6=v. (2.23) It follows from (2.1), (2.22) and (2.23) that

0≤

Z

R^N

|∇ξ_n,2|²− ^µ

|x|^2ξ

n,22 +V(x)η_n,2(x)ξ²_n,2

dx

≤

Z

R^N

|∇ξn,2|²− ^µ

|x|^2ηn,2(x)ξ²_n,2+V(x)ηn,2(x)ξ²_n,2

dx

→0.

Then, we have Z

R^N

|∇ξ_n,2|²− ^µ

|x|^2ξ

2n,2

dx→0, Z

R^NV(x)η_n,2(x)ξ²_n,2dx→0,

and _Z

R^NV(x)ξ²_n,2dx→1.

By a similar fashion as (2.19) and (2.20), we can conclude a contradiction.

On the other hand, by(h₂),(h₃),(h₈),(h₁₁),(F₀)and (1.5), there isC₁₇>0 such that

Z

R^N(f(x,h(v_n))h⁰(v_n)− f(x,h(v))h⁰(v))(v_n−v)dx

≤

Z

R^NC₁₇(|v_n|+|v_n|^2p−1+|v|+|v|^p2−1)|v_n−v|dx

≤C₁₇(kvnk₂+kvk₂)kvn−vk₂+

kvnk

p−2 p2 2

+kvk

p−2 p2 2

kvn−vk^p

2

=o(1).

(2.24)

Therefore, by (2.21) and (2.24), we have o(1) =hJ⁰(vn)−J⁰(v),vn−vi

=

Z

R^N

|∇(vn−v)|²+ (V(x)− ^µ

|x|²)(h(vn)h⁰(vn)−h(v)h⁰(v))(vn−v)

dx

−

Z

R^N(f(x,h(v_n))h⁰(v_n)− f(x,h(v))h⁰(v))(v_n−v)dx

≥C₁₅kv_n−vk+o(1).

This implies thatkvn−vk →0 asn→∞. Thus, the proof is complete.

To prove our main result in this paper, we need the following lemma.

Lemma 2.8(Symmetric Mountain Pass Theorem [26]). Let X be an infinite dimensional Banach space, X = Y^LZ, where Y is finite dimensional. If Ψ ∈ C¹(X,R)satisfies (C)_c-condition for all c>0, and

(I₁) _Ψ(₀) =0,Ψ(−u) =u for all u∈X;

(I₂) there exist constantsρ,α>₀such thatΨ |_∂Bρ∩Z≥α;

(I3) for any finite dimensional subspaceX˜ ⊂ X, there is R = R(X^˜) > 0 such that Ψ(u) ≤ 0 on X˜ \B_R;

thenΨ possesses an unbounded sequence of critical values.

3 Proof of Theorem 1.1

Let{e_i}is a total orthonormal basis of Eand defineX_i =_Rei, thenE=^L∞_i=1X_i. Let Y_j =

i

M

i=1

X_i, Z_j =

M∞ j+1

X_i, j∈_Z,

then E = Y_j^LZ_j and Y_j is finite-dimensional. Similar to Lemma 3.8 in [36], we have the following lemma.

Lemma 3.1([36]). Under assumptions(V₁)and(V₂), for2≤r <2^∗, β_j(r):= sup

u∈Z_j,kvk=1

kvk_r→0, j→_∞.

Before going further, we need to show that there existsC₁₈ >0 such that Z

R^N

|∇v|²− ^µ

|x|^2h

2(v) +V(x)h²(v)

dx≥ C₁₈kvk², ∀v∈Sρ, (3.1) where Sρ = {v ∈ E : kvk = ρ}. Indeed, by a similar argument as (2.16), we can get this conclusion. Moreover, by Lemma3.1, we can choose an integerκ≥1 such that

kvk²₂≤ ^C18

4c₁kvk, kvk

p 2p 2

≤ ^C18

4c₂kvk^p2, ∀v∈ Zκ. (3.2) Lemma 3.2. Assume that (V₁), (V₂) and (F₀) hold, then there exist constants ρ, α > 0 such that J|_Sρ∩Zκ ≥α.

Proof. For anyv∈Z_κ withkvk=ρ<1, by(h₃),(h₈), (3.1) and (3.2), we have J(v) = ¹

2 Z

R^N

|∇v|²− ^µ

|x|^2h

2(v) +V(x)h²(v)

dx−

Z

R^NF(x,h(v))dx

≥ ^C18

2 kvk²−

Z

R^N(c₁|h(v)|²+c₂|h(v)|^p)dx

≥ ^C18

2 kvk²−

Z

R^N(c₁|v|²+c₂|v|^p2)dx

≥ ^C18

2 kvk²− ^C18

4 kvk²−^C18 4 kvk^p2

= ^C18 4 kvk²

1− kvk^p−24

>_0.

since p ∈(4, 22^∗). This completes the proof.

Lemma 3.3. Assume that(V₁),(V₂),(F₀)and(F₁)hold, for any finite dimensional subspace E˜ ⊂E, there is R= R(E^˜)>₀such that

J(v)≤0, ∀v∈ E^˜ \B_R.

Proof. For any finite dimensional subspace ˜E ⊂ E, there is a positive integral numberκ such that ˜E ⊂ Y_κ. Suppose to the contrary that there is a sequence{v_n} ⊂ E^˜ such that kv_nk → _∞ and J(vn)>0. Hence

1 2

Z

R^N

|∇v_n|²− ^µ

|x|^2h

2(v_n) +V(x)h²(v_n)

dx>

Z

R^NF(x,h(v_n))dx. (3.3) Jointly with(h₃)_{, we have}

R

R^N F(x,h(v_n))dx kvnk² < ¹

2. (3.4)

Setη_n= _k^v_vⁿ

nk. Then up to a subsequence, we can assume that η_n *η in E,

η_n →η in L^r(_R^N) _{for 2}≤r <₂^∗ and

η_n →η a.e. on R^N.

SetA₁ ={x ∈_R^N :η(x)6=0}and A₂={x∈_R^N :η(x) =0}. If meas{A₁}>0, then by(F₁), (h5)and Fatou’s Lemma, we have

Z

A1

F(x,h(v_n)) kv_nk^{2 dx}=

Z

A1

F(x,h(v_n)) h⁴(v_n)

h⁴(v_n)

v²_n η_n²dx→_∞.

By(F₀)and(F₁), there existsC₁₉ >0 such that

F(x,t)≥ −C₁₉t², ∀(x,t)∈_R^N×_R.

Hence

Z

A2

F(x,h(vn))

kv_nk^{2 dx}≥ −C₁₉ Z

A2

h²(vn)

kv_nk^2dx≥ −C₁₉ Z

A2

η_n²dx.

Sinceη_n →ηinL²(_R^N), it is clear that lim inf

n→_∞ Z

A₂

F(x,h(v_n))

kv_nk^{2 dx}=0.

Consequently,

nlim→_∞ Z

R^N

F(x,h(vn))

kv_nk^{2 dx} =_∞.

By (3.4) we obtain ¹₂ >∞, a contradiction. This shows meas{A₁}=0 i.e. η(x) =0 a.e. onR^N. By the equivalency of all norms in ˜E, there existsC>0 such that

kvk²₂ ≥Ckvk², ∀v∈ E.^˜ Hence

0= lim

n→_∞kη_nk²₂ ≥C lim

n→+_∞kη_nk²=C, a contradiction. This completes the proof.

Now, we prove our main result.

Proof of Theorem1.1. Let Ψ = J, X = E,Y =Yκ and Z = Zκ. Obviously, J(0) =0 and (F₃) implies that J is even. By Lemma 2.7, 3.2 and Lemma 3.3, all conditions of Lemma 2.5 are satisfied. Thus, problem (2.6) has infinitely many nontrivial solutions sequence{vn}such that J(v_n)→_∞asn→∞. Namely, problem (1.1) also has infinitely many solutions sequence{u_n} such thatI(u_n)→_∞asn →_∞.

Acknowledgements

This work is supported in part by National Natural Science Foundation of China (11001274).

And the authors are grateful to Prof. Zhouxin Li for the reading of the paper and giving some helpful suggestions.

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