Infinitely many solutions for

Infinitely many solutions for

Schrödinger–Kirchhoff-type equations involving indefinite potential

Qingye Zhang

and Bin Xu

1College of Mathematics and Information Science, Jiangxi Normal University Nanchang 330022, PR China

2School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, PR China

Received 17 April 2017, appeared 16 August 2017 Communicated by Dimitri Mugnai

Abstract. In this paper, we study the multiplicity of solutions for the following Schrödinger–Kirchhoff-type equation

(− a+bR

R^N|∇u|²dx

4u+V(x)u= f(x,u) +g(x,u), x ∈_R^N, u∈H¹(R^N),

whereN≥3,a,b>0 are constants and the potentialVmay be unbounded from below.

Under some mild conditions on the nonlinearities f and g, we obtain the existence of infinitely many solutions for this problem. Recent results from the literature are generalized and significantly improved.

Keywords: Schrödinger–Kirchhoff-type equation, symmetric mountain pass lemma, variational method.

2010 Mathematics Subject Classification: 35J20, 35J60, 35J10.

1 Introduction and main results

In this paper, we consider the following Schrödinger–Kirchhoff-type equation (− a+_bR

R^N|∇u|²dx

4u+_V(_x)_u= _f(_x,u) +_g(_x,u)_{, x}∈_R^N,

u∈ H¹(_R^N), (1.1)

where N ≥ 3 and a,b > 0 are constants. If in (1.1), we set V(x) ≡ 0 and replace R^N by a smooth bounded domainΩ, then (1.1) reduces to the following Dirichlet problem

(− a+bR

Ω|∇u|²dx

4u= f(x,u), x∈_Ω,

u=0, x∈_∂Ω. ^(1.2)

BCorresponding author. Email: qingyezhang@gmail.com

Problem (1.2) is related to the stationary analogue of the Kirchhoff equation utt−

a+b

Z

Ω|∇u|²dx

4u= f(x,u),

which was presented by Kirchhoff in 1883 [8] as a generalization of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. In [11], Lions first introduced an abstract functional analysis framework to this model. After that, problems like type (1.2) have been studied by many authors, see [3,4,15,16,20,25,27] and the references therein.

More recently, with the aid of variational methods, the existence and multiplicity of various solutions for equations of type (1.1) have also been extensively investigated in the literature, see, for instance, [1,2,5,6,9,10,12,21–24,26,29] and the references therein. Here we emphasize that almost in all these mentioned papers the conditions imposed on the potentialV always imply thatV is bounded from below, which is crucial for the corresponding results.

In the present paper, different from the references mentioned above, we are going to study the existence of infinitely many solutions for (1.1) in the case where the potential V may be unbounded from below. Specifically, we first assume thatVsatisfies

(S₁) V∈ L^q_loc(_R^N)andV⁻:=min{V, 0} ∈L^∞(_R^N) +L^q(_R^N)for someq∈[2,∞)∩(^N₂,∞). This type of assumptions on the potential V has already been introduced in [13] to study Schrödinger equations (see also [28]), which ensures that the Schrödinger operator S :=

−_a∆+V, defined as a form sum, is self-adjoint and semibounded on L²(_R^N) (see Theo- rem A.2.7 in [19]). We denote by σ(S)⊂ _Rthe spectrum, σ_ess(S)the essential spectrum and σ_pp(S)the pure point spectrum of S respectively.

Consider the nondecreasing sequence of min-max values defined by λ_k = _inf

U∈U_k

sup

u∈U\{0}

R

R^N a|∇u|²+V(x)u² dx R

R^Nu²dx , ∀k∈_N,

where U_k is the family of all k-dimensional subspaces of C₀^∞(_R^N). It is known that λ_∞ := lim_k→_∞λ_k = _infσ_ess(S). Moreover, λ_k ∈ σ_pp(S) _whenever λ_k < λ_∞ (cf. [17,18] for details).

Then we make the further assumption onV.

(S₂) λ_∞ >0.

For the nonlinearities, we present the following assumptions.

(S₃) The function f ∈ C(_R^N ×_R,R) is odd in u, and there exist constants ν ∈ (1, 2) and µ∈ (2^∗/(2^∗−ν), 2/(2−ν)]and a nonnegative functionξ ∈ L^µ(_R^N)such that

|f(x,u)| ≤ξ(x)|u|^ν−1, ∀(x,u)∈ _R^N×_R, where 2^∗ :=2N/(N−2)is the critical exponent.

(S₄) There exist anx₀ ∈_R^N and a constantr₀>0 such that lim inf

u→0 inf

x∈Br0(x0)u⁻²F(x,u)

!

>−_∞,

and

lim sup

u→0

x∈Binfr0(x0)u⁻²F(x,u)

!

= +_∞

where B_r0(x₀)is the ball inR^N centered at x₀ with radiusr₀ and F(x,u):=

Z _u

0 f(x,t)dt.

(S₅) g∈C(_R^N×_R,R)is odd inu, and there existsd∈(0,λ_∞)such that

|g(x,u)| ≤d|u|, ∀(x,u)∈_R^N×_R.

Our main result reads as follows.

Theorem 1.1. Suppose that (S₁)–(S5) are satisfied. Then (1.1) possesses a sequence of nontrivial solutions{u_k}_k∈N⊂ H¹(_R^N)with u_k →0in H¹(_R^N)as k→_∞.

Remark 1.2. In Theorem 1.1, the potential V satisfying (S₁) and (S2) may not be coercive or bounded from below. Moreover, the nonlinear term f satisfying (S₃) and (S₄) may be partially oscillatory near the origin. This is in sharp contrast with the aforementioned references. To the best of our knowledge, there is little literature concerning infinitely many solutions for (1.1) in this situation. In fact, it is easy to see that conditions (S₁) and (S₂) are rather weaker than the usual one in the existing literature that the potentialV ∈C(_R^N)with lim_|x|→∞V(x) = +_∞or inf_x∈RNV(x)>0.

Remark 1.3. Theorem 1.1 also essentially improves some related results in the existing liter- ature. Compared to Theorem 6 in [26], our conditions (S₁) and (S₂) on the potential V are weaker than (V₁) there, and our conditions (S₃) and (S₄) on the nonlinear term f are much weaker than (f5) there if we just take g = 0 in (1.1). In fact, there are many functions V and f which satisfy our conditions (S₁)–(S₄) but do not satisfy the condition (V₁) and (f₅) in [26].

For instance, let

V(x) =V0(x) +V,

where V₀ ∈ L^q(_R^N) for some q ≥ 2 is a given non-positive function and unbounded from below. Then it is evident thatVsatisfies (S₁) and (S2) if the positive constantVis chosen to be large enough. Moreover,Vis also unbounded from below. In addition, let

F(x,u) =

(e^−|x|2|u|^αsin²(|u|^−e), ∀x ∈_R^N, 0< |u|<π^−1/e, 0, ∀x ∈_R^N, u=0 or|u| ≥π^−1/e

be the primitive function of f with respect tou, wheree>0 is small enough andα∈(1+e, 2). Then it is easy to check that f satisfies conditions (S₃) and (S₄) with ν = α−e and ξ(x) = (α+e)e^−|x|2.

2 Notations and preliminaries

Throughout this paper, we always use the following notations:

• H¹(_R^N)is the usual Sobolev space equipped with the standard norm kuk²_H1 =

Z

R^N |∇u|²+u² dx, andH⁻¹(_R^N)is the dual space ofH¹(_R^N).

• D^1,2(_R^N)is the completion ofC₀^∞(_R^N)with respect to the norm kuk²_D1,2 =

Z

R^N|∇u|²dx.

• L^p(_Ω), 1 ≤ p ≤ _∞,Ω ⊆ _R^N, denotes a Lebesgue space, and the norm in L^p(_Ω) is denoted bykuk_p,ΩwhenΩis a proper subset ofR^N, bykuk_p whenΩ=_R^N.

• For any R>0,B_R denotes the ball inR^N centered at 0 with radius R.

• →(resp.*) denotes the strong (resp. weak) convergence.

In what follows it will always be assumed that (S₁) and (S₂) are satisfied. As pointed out in [13], the form domain of the Schrödinger operatorS is

E:=

u∈ H¹(_R^N)|

Z

R^N a|∇u|²+V(x)u²

dx<_∞

, which becomes a Hilbert space if it is equipped with the inner product

(u,v)₀ :=

Z

R^N(a∇u· ∇v+V(x)uv+l₀uv)dx, ∀u,v∈ E,

wherel0 > −infσ(S) = −λ₁ is a fixed positive constant. We denote by k · k₀ the associated norm.

Lemma 2.1. E is continuously embedded into H¹(_R^N), that is, kuk_H1 ≤ c₀kuk₀, ∀u∈E for some c0>0.

Proof. Arguing indirectly, we assume that there exists a sequence{u_n}_n∈N⊂Esuch that kunk²_H1 =

Z

R^N |∇un|²+u²_n

dx≡1, ∀n ∈_N (2.1)

and

kunk²₀=

Z

R^N a|∇un|²+V(x)u²_n+l0u²_n

dx→0 asn→_∞. (2.2)

Sincel₀ >−infσ(S), then it holds that Z

R^N a|∇un|²+V(x)u²_n+l0u²_n

dx≥c₁ Z

R^Nu²_ndx (2.3)

for somec₁ >0. By (2.2) and (2.3), we get ku_nk₂ =

_Z

R^Nu²_ndx 1/2

→0 asn→_∞. (2.4)

Let

V⁻=V₁⁻+V₂⁻

with V₁⁻ ∈ L^∞(_R^N) andV₂⁻ ∈ L^q(_R^N), where V⁻ and qare given in (S₁). Combining (2.1), (2.4), Hölder’s inequality and the Gagliardo–Nirenberg inequality, we have

Z

R^NV⁻u²_ndx

= Z

R^NV₁⁻u²_ndx+

Z

R^NV₂⁻u²_ndx

≤

Z

R^N|V₁⁻u²_n|dx+

Z

R^N|V₂⁻u²_n|dx

≤ kV₁⁻k_∞ku_nk²₂+kV₂⁻k_qku_nk²_2q/(q−1)

≤ kV₁⁻k_∞ku_nk²₂+c₂kV₂⁻k_qk∇u_nk^N/q₂ ku_nk⁽₂^2q−N)/q →0 asn→_∞, wherec₂ >0 is a constant depending onq. This together with (2.2) and (2.4) yields

Z

R^N |∇un|²+u²_n

dx→0 as n→_∞, which contradicts (2.1). The proof is completed.

For later use, we introduce the new inner product in E as follows. Choose ¯d ∈ (_d,λ_∞) such that ¯d 6= λ_k for all k ∈ _{N, where} d is the constant given in (S₅). Denote by λ_k0 the first eigenvalue of the Schrödinger operator S greater than ¯d. Let E⁻ be the subspace of E spanned by the eigenfunctions with corresponding eigenvalues less than ¯d. Note the fact that λ_∞ = lim_k→_∞λ_k and λ_k ∈ σpp(S)whenever λ_k < λ_∞. Then it is evident that E⁻ is a finite dimensional subspace of E. If there is no eigenvalue of the Schrödinger operator S greater than ¯d, then we set λ_k0 = λ_∞ and E⁻ is empty in this case. Let E⁺ be the orthogonal com- plement of E⁻ inE with respect to the inner product(·,·)₀. ThenEpossesses the orthogonal decompositionE=E⁻⊕E⁺. By definition, it holds that

Z

R^N a|∇u|²+V(x)u²

dx ≥λ_k0 Z

R^Nu²dx, ∀u∈E⁺. (2.5) Now we can define the new inner product(·,·)and the induced normk · kinEby

(u,v) =

Z

R^N a∇u⁺· ∇v⁺+V(x)u⁺v⁺−du^{¯ +}v⁺ dx

−

Z

R^N a∇u⁻· ∇v⁻+V(x)u⁻v⁻−du^{¯ −}v⁻ dx,

(2.6)

kuk= q

(u,u) (2.7)

for all u = u⁻+u⁺, v = v⁻+v⁺ ∈ E with u^±, v^± ∈ E^±. Note the fact that E⁻ and E⁺ are also orthogonal with respect to the usual inner product in L²(_R^N). Then it is evident that Epossesses the same orthogonal decomposition E = E⁻⊕E⁺ with respect to the new inner product(·,·). Moreover, we have

Z

R^N a|∇u|²+V(x)u²−du^{¯ 2}

dx =ku⁺k²− ku⁻k² (2.8) for all u=u⁻+u⁺∈Ewith u^±∈ E^±.

Lemma 2.2. The normsk · kandk · k₀ are equivalent in E.

Proof. It suffices to show thatk · kandk · k₀are equivalent inE⁺sinceE⁻is finite dimensional.

On the one hand, by (2.8), there holds kuk²=

Z

R^N a|∇u|²+V(x)u²−du^{¯ 2} dx

≤

Z

R^N a|∇u|²+V(x)u²+l₀u²

dx=kuk²₀, ∀u∈E⁺.

(2.9)

On the other hand, invoking (2.5) and (2.8), we get kuk² =

Z

R^N a|∇u|²+V(x)u²−du^{¯ 2} dx

≥ ^λk0 −d^¯ λ_k0

Z

R^N a|∇u|²+V(x)u² dx

≥ ^λk0 −d^¯ 2λ_k0

Z

R^N a|∇u|²+V(x)u²

dx+ ^λk0−d^¯ 2l0

Z

R^Nl₀u²dx

≥c₃ Z

R^N a|∇u|²+V(x)u²+l₀u² dx

=c₃kuk²₀, ∀u∈E⁺,

(2.10)

where c₃ = _min{(λ_k0−d^¯)_/2λk0,(λ_k0−d^¯)/2l₀} > 0 by the choice of ¯d and λ_k0. Combining (2.9) and (2.10), we know thatk · kandk · k₀are equivalent inE⁺. The proof is completed.

Hereafter, we always use the inner product(·,·)and the induced normk · kinE. Moreover, we write E^∗ for the dual space of E, and h·,·i : E^∗×E → _R for the dual pairing. From Lemma 2.1 and Lemma 2.2, we immediately know that E is continuously embedded into H¹(_R^N). Furthermore, using the Sobolev embedding theorem, we also get the following lemma.

Lemma 2.3. E is continuously embedded into D^1,2(_R^N)and L^p(_R^N)for all p ∈ [2, 2^∗], and hence there exist constants c₄, τp >0such that

kuk_D1,2 ≤c₄kuk, ∀u∈E (2.11) and

kuk_p≤ τ_pkuk, ∀u∈ E and p∈[2, 2^∗]. (2.12) Moreover, for any bounded domainΩ⊂_R^N, E is compactly embedded into L^p(_Ω)for all p∈[1, 2^∗).

3 Variational setting and proof of the main result

In this section, we will first introduce the variational setting for (1.1). To this end, we define functionalsΨ_i(i=1, 2, 3)andΦonEby

Ψ₁(u) = ^b 4

_Z

R^N|∇u|²dx 2

, Ψ₂(u) =

Z

R^NF(x,u)dx, Ψ₃(u) =

Z

R^N

d¯

2u²−G(x,u)

dx

and

Φ(u) = ¹ 2

Z

R^N a|∇u|²+V(x)u²

dx+ ^b 4

_Z

R^N|∇u|²dx 2

−

Z

R^N(F(x,u) +G(x,u))dx

= ¹ 2

Z

R^N a|∇u|²+V(x)u²−du^{¯ 2}

dx+ ^b 4

_Z

R^N|∇u|²dx 2

−

Z

R^N F(x,u)dx +

Z

R^N

d¯

2u²−G(x,u)

dx

= ¹

2ku⁺k²−¹

2ku⁻k²+Ψ₁(u)−Ψ2(u) +Ψ3(u)

(3.1)

for all u = u⁻+u⁺ ∈ E with u^± ∈ E^±. Here ¯d is the constant in (2.8) and G(x,u) := Ru

0 g(x,t)dtis the primitive function of g(x,u)with respect tou.

Proposition 3.1. Assume that (S₁)–(S₃) and(S₅) are satisfied. Then Ψ_i ∈ C¹(E,R) for i = 1, 2, 3 withΨ_i⁰ :E→E^∗being completely continuous for i=2, 3, and henceΦ∈C¹(E,R). Moreover,

hΨ₁⁰(u),vi=b _Z

R^N|∇u|²dx _Z

R^N∇u· ∇vdx, (3.2)

hΨ₂⁰(u),vi=

Z

R^N f(x,u)vdx, (3.3)

hΨ₃⁰(u),vi=

Z

R^N(du^¯ −g(x,u))vdx, (3.4)

hΦ⁰(u),vi= (u⁺,v⁺)−(u⁻,v⁻) +hΨ⁰₁(u),vi − hΨ⁰₂(u),vi+hΨ⁰₃(u),vi

=

a+b Z

R^N|∇u|²dx _Z

R^N∇u· ∇vdx+

Z

R^NV(x)uvdx

−

Z

R^N(f(x,u) +g(x,u))vdx

(3.5)

for all u = u⁻+u⁺, v = v⁻+v⁺ ∈ E with u^±, v^± ∈ E^±. In addition, if u ∈ E ⊆ H¹(_R^N)is a critical point ofΦon E, then it is a solution of(1.1).

Proof. First, we show thatΨ₁∈ C¹(E,R)and (3.2) holds. Define a functionalΨ₀on D^1,2(_R^N) by

Ψ0(u) = ^b 4

Z

R^N|∇u|²dx 2

. Evidently,Ψ0 ∈C¹(D^1,2(_R^N),R)and

hΨ₀⁰(u),vi=b _Z

R^N|∇u|²dx _Z

R^N∇u· ∇vdx, ∀u, v∈ D^1,2(_R^N). (3.6) Let ι : E → D^1,2(_R^N) be the continuous embedding in Lemma 2.3. Since Ψ₁ = Ψ₀◦_{ι, we} immediately know by (3.6) thatΨ₁∈ C¹(E,R)and (3.2) holds.

Next, we verify (3.3) by definition and prove that Ψ₂ ∈ C¹(E,R)with Ψ₂⁰ : E → E^∗ being completely continuous. By (S3), there holds

|F(x,u)| ≤ν⁻¹ξ(x)|u|^ν, ∀(x,u)∈_R^N×_R. (3.7) For notational simplicity, we set

µ^∗ := ^µν

µ−1. (3.8)

Since ν ∈ (1, 2) and µ ∈ (2^∗/(2^∗−ν), 2/(2−ν)] in (S₃), we get µ^∗ ∈ [2, 2^∗). Then for any u∈E, by (2.12), (3.7) and Hölder’s inequality, we have

Z

R^N|F(x,u)|dx≤

Z

R^Nν⁻¹ξ(x)|u|^νdx

≤ν⁻¹kξk_µkuk^ν_µ∗

≤ν⁻¹τ_µ^ν∗kξk_µkuk^ν <_∞,

(3.9)

whereτ_µ^∗ is the constant given in (2.12). ThusΨ₂is well defined. For any givenu∈ E, define an associated linear operatorJ(u):E→_Ras follows:

hJ(u),vi=

Z

R^N f(x,u)vdx, ∀v∈ E. (3.10) By (S₃), (2.12) and Hölder’s inequality, there holds

|hJ(u),vi| ≤

Z

R^Nξ(x)|u|^ν−1|v|dx

≤ kξk_µkuk^ν_µ⁻∗¹kvk_µ^∗

≤τ_µ^ν∗kξk_µkuk^ν−1kvk_, ∀v∈E,

(3.11)

which shows thatJ(u)is well defined and bounded. On the other hand, it follows from (S3) that

|f(x,u+ηv)v| ≤2^ν−1ξ(x)(|u|^ν−1|v|+|v|^ν), ∀x∈_R^N, η∈ [0, 1]andu,v∈_R. (3.12) Then for anyu,v ∈ E, combining (3.9)-(3.12), the mean value theorem and Lebesgue’s domi- nated convergence theorem, we have

limt→0

Ψ₂(u+tv)−Ψ₂(u)

t =lim

t→0

Z

R^N f(x,u+θ(x)tv)vdx

=

Z

R^N f(x,u)vdx

=hJ(u),vi,

(3.13)

whereθ(x)∈ [0, 1] depends onu,v,t. This shows that Ψ₂ is Gâteaux differentiable on Eand the Gâteaux derivative ofΨ₂atuisJ(u).

In order to prove thatΨ₂ ∈C¹(E,R)_andΨ⁰₂:E→ E^∗ is completely continuous, it suffices to prove thatJ :E→ E^∗ is completely continuous. To this end, we claim that ifun *uin E, then for anyR>0,

Z

BR

|f(x,un)− f(x,u)|^p0dx→0 asn→_∞, (3.14) where p₀ := max{2^∗/(2^∗−1),µ/(µ(ν−1) +1)} with µ and ν given in (S₃). Arguing in- directly, by Lemma 2.3, we assume that there exist constants R₀,ε₀ > 0 and a subsequence {u_nk}_k∈Nsuch that

u_nk →uin L^p∗0(B_R0)andu_nk →ua.e. inB_R0 ask→_∞ (3.15)

but _Z

BR0

|f(x,un_k)− f(x,u)|^p0dx ≥ε₀, ∀k∈_N, (3.16)

where p^∗₀ := p₀µ(ν−1)/(µ−p₀) ∈ [1, 2^∗) by (S₃) and the choice of p₀ above. Due to (3.15), passing to a subsequence if necessary, we can further assume that

∑

kun_k−uk_p^∗

0,BR0 <+_∞.

Let w(x) = _∑^∞_k=1|u_nk(x)−u(x)| for all x ∈ B_R0, then w ∈ L^p∗0(B_R0). By virtue of (S₃) and Hölder’s inequality, we get

|f(x,un_k)− f(x,u)|^p0

≤ (|f(x,u_nk)|+|f(x,u)|)^p0

≤ ξ(x)^p0|un_k|^ν−1+|u|^ν−1p0

≤2^p0ξ(x)^p0|u_nk|^p0(ν−1)+|u|^p0(ν−1)

≤2^p0ν+1ξ(x)^p0|un_k−u|^p0(ν−1)+|u|^p0(ν−1)

≤2^p0ν+1ξ(x)^p0|w|^p0(ν−1)+|u|^p0(ν−1), ∀k∈_Nandx∈ B_R0

(3.17)

and Z

BR0

ξ(x)^p0|w|^p0(ν−1)+|u|^p0(ν−1)dx≤ kξk^p_µ⁰kwk^p_p⁰∗^(ν−1) 0,BR0

+kuk^p_p⁰∗^(ν−1) 0,BR0

<+_{∞. (3.18)} Combining (3.15), (3.17), (3.18) and Lebesgue’s dominated convergence theorem, we have

klim→_∞ Z

BR0

|f(x,un_k)− f(x,u)|^p0dx=0, which contradicts (3.16). Thus the claim is true.

Now let u_n * u in E as n → _{∞, then} {u_n} is bounded in E and hence there exists a constant D₀ >0 such that

ku_nk^ν+ku_nkkuk^ν−1≤ D₀, ∀n∈_N. (3.19) For anye>0, by (S3), there exists Re>0 such that

Z

R^N\BRe

ξ(_x)^µ_dx 1/µ

< ^e

2D₀τ_µ^ν∗

. (3.20)

Combining (S₃), (3.19), (3.20) and Hölder’s inequality, we have Z

R^N\BRe

|f(x,un)− f(x,u)||v|dx≤

Z

R^N\BRe

(|f(x,un)|+|f(x,u)|)|v|dx

≤

Z

R^N\BRe

ξ(x)|un|^ν−1+|u|^ν−1|v|dx

≤ _Z

R^N\BRe

ξ(x)^µdx 1/µ

kunk^ν_µ⁻∗¹+kuk^ν_µ⁻∗¹

kvk_µ^∗

≤τ_µ^ν∗ _Z

R^N\BRe

ξ(x)^µdx 1/µ

ku_nk^ν−1+kuk^ν−1

< ^e

2, ∀n∈_Nandkvk=1.

(3.21)

For theR_egiven in (3.20), by Hölder’s inequality and (3.14), there exists N_e ∈_Nsuch that Z

B_Re

f(x,u_n)− f(x,u)|v|dx≤ _Z

B_Re

f(x,u_n)− f(x,u)

p0

dx 1/p₀

kvk_p¯0,BR

e

≤ Z

BRe

f(x,u_n)− f(x,u)

p0

dx 1/p0

kvk_p¯0

≤τp¯₀

Z

BRe

f(x,un)− f(x,u)

p0

dx 1/p0

< ^e

2, ∀n≥ N_e andkvk=1,

(3.22)

where ¯p₀:= p₀/(p₀−1)∈(1, 2^∗]and p₀is the constant given in (3.14), andτp¯₀ is the constant given in (2.12). Combining (3.21) and (3.22), we have

kJ(un)− J(u)k_E∗ = sup

kvk=1

|hJ(un)− J(u),vi|

= sup

kvk=1

Z

R^N(f(x,un)− f(x,u))vdx

≤ sup

kvk=1

Z

BRe

f(x,un)− f(x,u)|v|dx + sup

kvk=1

Z

R^N\B_Re

f(x,u_n)− f(x,u)|v|dx

≤ ^e 2 +^e

2 =e, ∀n≥ Ne. This shows thatJ :E→E^∗ is completely continuous.

Then, taking (S₅) into account and using similar arguments to those above, one can also prove thatΨ₃ ∈ C¹(E,R)with Ψ⁰₃ :E→ E^∗ being completely continuous and (3.4) holds. For simplicity, we omit the proof here.

Finally, combining (2.6) and (3.1)–(3.4), we immediately know thatΦ∈C¹(E,R)and (3.5) holds. In addition, it is known that any critical pointu∈ E⊆ H¹(_R^N)of the functionalΦis a solution of (1.1). The proof is completed.

We will use the following variant symmetric mountain pass lemma due to [7] (see also [14]) to prove that (1.1) possesses a sequence of weak solutions. Before stating this theorem, we first recall the notion of genus.

Let Ebe a Banach space and A a subset of E. Ais said to be symmetric ifu ∈ A implies

−u ∈ A. Denote byΓthe family of all closed symmetric subset ofEwhich does not contain 0.

For any A ∈ Γ, define the genus γ(A) of A by the smallest integer k such that there exists an odd continuous mapping from A to R^k\ {0}. If there does not exist such a k, define γ(A) =∞. Moreover, setγ(_∅) =0. For eachk∈_{N, letΓk} = {A∈_Γ|γ(A)≥k}.

Theorem 3.2([7, Theorem 1]). Let E be an infinite dimensional Banach space andΦ∈C¹(E,R)an even functional withΦ(0) =0. Suppose thatΦsatisfies

(Φ1) Φis bounded from below and satisfies(PS)condition.

(Φ2) For each k∈N, there exists an Ak ∈_Γk such thatsup_u∈A

kΦ(u)<0.

Then either (i) or (ii) below holds.

(i) There exists a critical point sequence{u_k}such thatΦ(u_k)<0andlim_k→∞u_k =0.

(ii) There exist two critical point sequences {u_k} and {v_k} such that Φ(u_k) = 0, u_k 6= 0, lim_k→∞u_k =0,Φ(v_k)<0,lim_k→∞Φ(v_k) =0, and{v_k}converges to a non-zero limit.

In order to apply Theorem3.2, we will show in the following lemmas that the functionalΦ defined in (3.1) satisfies conditions (_Φ1)_{and (Φ2}) in Theorem3.2. The proof of these lemmas is partially motivated by [21] and [7].

Lemma 3.3. Let(S₁)–(S₃)and(S₅)be satisfied. ThenΦis coercive and bounded from below.

Proof. We first prove thatΦis coercive. Arguing indirectly, we assume that for some sequence {u_n}_n∈N ⊂ Ewithku_nk →∞, there is a constant M >0 such that Φ(u_n)≤ Mfor alln ∈ _N.

Let u_n = u⁻_n +u⁺_n with u^±_n ∈ E^±. If we setv_n = u_n/ku_nk_{for all} n ∈ _{N, then}kv_nk ≡ _{1, and} vn= v⁻_n +v⁺_n with v^±_n =u^±_n/kunk ∈E^±. Note thatE⁻is finite dimensional. Thus, passing to a subsequence if necessary, we can assume by Lemma2.3that

vn*v, v⁻_n →v⁻, v⁺_n *v⁺andvn →va.e. inR^N asn→_∞ (3.23) for somev=v⁻+v⁺ ∈Ewithv^±∈ E^±. By (S₅), there hold

0≤ ^d¯−d

2 u² ≤ ^d¯

2u²−G(x,u)≤ ^d¯+d

2 u², ∀(x,u)∈_R^N×_R (3.24) and

du¯ ²−g(x,u)u≥(d^¯−d)u²≥0, ∀(x,u)∈ _R^N×_R (3.25) since ¯dis chosen to be greater thandin Section 2. Combining (3.1), (3.9) and (3.24), we have

M≥Φ(u_n)≥ ¹

2ku⁺_nk²− ¹

2ku⁻_nk²−

Z

R^N|F(x,u_n)|dx

≥ ¹

2ku⁺_nk²− ¹

2ku⁻_nk²−ν⁻¹τ_µ^ν∗kξk_µkunk^ν, ∀n∈_N.

(3.26)

Multiplying both sides of (3.26) bykunk⁻², we get

kv⁺_nk²≤ kv⁻_nk²+o(1) asn→_∞ (3.27) since ν<2 in (S3) andkunk →_∞.

Ifv = 0, then v⁻_n → 0 and hence v⁺_n → 0 by (3.27). This implies v_n → 0, which leads to a contradiction since kvnk ≡ 1. Therefore, v 6= 0. Note that vn * v in D^1,2(_R^N) since E is continuously embedded intoD^1,2(_R^N). Then it follows from the weak lower semi-continuity of the norm k · k_D1,2 in D^1,2(_R^N)that

lim inf

n→_∞

"

b 4ku_nk⁴

Z

R^N|∇un|²dx 2#

=lim inf

n→_∞

"

b 4

Z

R^N|∇vn|²dx 2#

=lim inf

n→_∞

b

4kv_nk⁴_D1,2

= ^b 4

lim inf

n→_∞ kvnk_D1,2

4

≥ ^b

4kvk⁴_D1,2 >0.

(3.28)

Combining (3.1), (3.9) and (3.24), we have M≥Φ(un)≥ ¹

2ku⁺_nk²− ¹

2ku⁻_nk²+ ^b 4

_Z

R^N|∇un|²dx 2

−

Z

R^N|F(x,un)|dx

≥ ¹

2ku⁺_nk²− ¹

2ku⁻_nk²+ ^b 4

Z

R^N|∇un|²dx 2

−ν⁻¹τ_µ^ν∗kξk_µkunk^ν, or equivalently,

b 4

_Z

R^N|∇u_n|²dx ₂

≤ ¹

2ku⁻_nk²−¹

2ku⁺_nk²+ν⁻¹τ_µ^ν∗kξk_µku_nk^ν+M, (3.29) where µ^∗ and τ_µ^∗ are the constants given in (3.8) and (2.12) respectively. Multiplying both sides of (3.29) byku_nk⁻⁴and lettingn→_{∞, we get}

lim inf

n→_∞

"

b 4ku_nk⁴

_Z

R^N|∇u_n|²dx 2#

≤0, which contradicts (3.28). Therefore,Φis coercive.

Next, we show that Φ is bounded from below. Combining (2.11), (2.12), (3.1), (3.9) and (3.24), we have

|Φ(u)| ≤ ¹

2kuk²+^b 4

Z

R^N|∇u|²dx 2

+

Z

R^N|F(x,u)|dx+

Z

R^N

d¯

2u²−G(x,u)

dx

≤ ¹

2kuk²+^bc

44

4 kuk⁴+ν⁻¹τ_µ^ν∗kξk_µkuk^ν+ ^d¯+d

2 τ₂²kuk²,

(3.30)

where c₄ andτ₂ are the constants given in (2.11) and (2.12) respectively. This implies thatΦ maps bounded sets inEinto bounded sets inR. Then it follows from the coercivity thatΦis bounded from below. The proof is completed.

Lemma 3.4. Assume that(S₁)–(S3)and(S5)are satisfied. ThenΦsatisfies(PS)condition.

Proof. Let{u_n}_n∈N⊂Ebe a (PS)-sequence, i.e.,

|Φ(u_n)| ≤D₁ and Φ⁰(u_n)→0 as n→_∞ (3.31) for someD₁>0. Note first thatΦis coercive by Lemma3.3. This together with (3.31) implies that{u_n}_n∈N is bounded inE. Thus there exists a subsequence{u_nk}_k∈Nsuch that

un_k *u0 ask →_∞ (3.32)

for someu₀∈ E. Let

u_nk =u⁻_nk+u⁺_nk and u₀ =u₀⁻+u⁺₀ withu^±_nk, u^±₀ ∈ E^±. SinceE⁻ is finite dimensional, we get

u⁻_nk →u⁻₀ and u⁺_nk *u⁺₀ ask→_{∞. (3.33)}