J. Math. Anal. Appl. 399 (2013) 340–351

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Existence and multiplicity of semiclassical solutions for asymptoticallyHamiltonian elliptic systemsLu Xiao a, Jun Wang b,∗, Ming Fan a, Fubao Zhang c

a School of management, Jiangsu University, Zhenjiang, Jiangsu, 212013, PR Chinab Department of Mathematics, Jiangsu University, Zhenjiang, Jiangsu, 212013, PR Chinac Department of Mathematics, Southeast University, Nanjing 210096, PR China

a r t i c l e i n f o

Article history:Received 25 June 2012Available online 17 October 2012Submitted by Mr. V. Radulescu

Keywords:Variational methods(C)c -conditionGeneralized linking theorem

a b s t r a c t

This paper is concerned with the following nonperiodic Hamiltonian elliptic system−ε2u + V (x)u = Hu(x, u, v) in RN ,

−ε2v + V (x)v = −Hv(x, u, v) in RN ,u(x) → 0 and v(x) → 0 as |x| → ∞,

where ε > 0 is a small parameter, and the potentialV is bounded below, andH is asymptot-ically linear in z as |z| → ∞ with z = (u, v). By applying a generalized linking theorem ofstrongly indefinite functionals, we obtain the existence of multiple semiclassical solutionsfor the above system.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction and main results

In this paper we study the following nonperiodic elliptic system in Hamiltonian form:

(Pε)

−ε2u + V (x)u = Hu(x, u, v) in RN ,

−ε2v + V (x)v = −Hv(x, u, v) in RN ,u(x) → 0 and v(x) → 0 as |x| → ∞,

where N ≥ 1, V ∈ C(RN ,R) and H ∈ C1(RN× R2,R). Set J0 =

1 00 −1

. Thenthe system (Pε) can be written as

−ε2∆z + V (x)z = J0Hz(x, z), z(x) = (u(x), v(x)) → 0 as |x| → ∞,

which can be regarded as the stationary system of the nonlinear vector Schrödinger equation

ih∂φ

∂t= −

h2

2m∆φ + γ (x)φ − J0f (x, |φ|)φ,

where φ(x, t) = z(x)e−iEth , V (x) = γ (x) − E, ε2 =

h22m and Hz(x, z) = f (x, |z|)z. Set λ = ε−2. Then the system (Pε) is

equivalent to the following one

(Pλ)

−u + λV (x)u = λHu(x, u, v) in RN ,

−v + λV (x)v = −λHv(x, u, v) in RN ,u(x) → 0 and v(x) → 0 as |x| → ∞.

∗ Corresponding author.E-mail addresses: hnlulu@126.com (L. Xiao), wangmath2011@126.com (J. Wang), fanming@ujs.edu.cn (M. Fan), zhangfubao@seu.edu.cn (F. Zhang).

0022-247X/$ – see front matter© 2012 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2012.10.010

L. Xiao et al. / J. Math. Anal. Appl. 399 (2013) 340–351 341

If the parameter ε = 1 and Ω is a bounded domain, many mathematicians are devoted to study the existence andmultiplicity of solutions for the system (Pε) in recent years. For example, see [1–6] and the references therein. Benci andRabinowitz [3] first considered the system (Pε). By using the directmin–maxmethod, they obtained solutions for the system(Pε) in the space ofH1

0 (Ω)×H10 (Ω). Later, Hulshof and van der Vorst [2] also considered the system (Pε). Instead of working

in the space of H10 (Ω)× H1

0 (Ω), the authors used a suitable family of products of fractional Sobolev spaces, and turned outthat these kinds of spaces are the natural ones for this problem. By doing so, the authors also proved that the system (Pε) hasexistence of solutions. In [1] de Figueiredo and Felmer considered the super-quadratic case for the system (Pε). By applyinga critical point theorem, they acquired solutions for the system (Pε). In [6], Kryszewski and Szulkin considered the followingsystem

(X)

−u = Fv(x, u, v) inΩ ⊂ RN ,−v = Fu(x, u, v) inΩ,u(x) = v(x) = 0 on ∂Ω.

By developing an infinite-dimensional Morse theory for strongly indefinite functionals, for N ≥ 1, they obtained one weaksolution for the system (X). In particular, if N = 1, they proved that the system (X) has at least two nontrivial solutions.Recently, by developing a critical point theorem for strongly indefinite functionals, de Figueiredo and Ding [4] studied thesystem−u = Hu(x, u, v) inΩ ⊂ RN ,

−v = −Hv(x, u, v) inΩ,u(x) = v(x) = 0 on ∂Ω,

where H satisfies H(x, u, v) ∼ |u|p + |v|q + R(x, u, v)with lim|u|+|v|→∞R(x,u,v)|u|p+|v|q = 0, 1 < p < 2∗

=2NN−2 and q > 1. Under

some additional conditions on R, they proved the existence of multiple solutions for the above system. More recently, deFigueiredo et al. [5] have treated the following autonomous system via an Orlicz space approach

−u = g(v) u(x) > 0 inΩ,−v = f (u) v(x) > 0 inΩ,u(x) = v(x) = 0 on ∂Ω.

Under some superlinear conditions on f and g , the authors obtained one solution for the system.Up till now, some authors considered the system (Pε) in the space of RN . For the parameter ε = 1, the papers [7–17]

proved the existence of multiple nontrivial solutions for the system (Pε). Most of them focused on the case V ≡ 1. Themaindifficulty of this problem is the lack of compactness for Sobolev’s embedding theorem. In the early results [8–11], the authorsmanaged to recover the compactness for Sobolev’s embedding, by working in the space of radially symmetric functions. Inthis way, they proved the existence of radial solutions for the system (Pε). Later, by using the dual variational method,the papers [12–15] also overcome the lack of compactness for Sobolev’s embedding theorem. Therefore, they obtainedsolutions for the system (Pε). Recently, in [18], Bartsch and Ding have developed a generalized linking theorem for thestrongly indefinite functionals (see [19,20] for related results), which provided another way to deal with such problems.The papers [7,16,21] considered the following system

(Z)

−u + V (x)u = Rv(x, u, v) in RN ,

−v + V (x)v = Ru(x, u, v) in RN ,u(x) → 0 and v(x) → 0 as |x| → ∞.

By applying the generalized linking theorems of [18], they proved the existence of solutions for the system (Z). Let Ω bea domain of RN , not necessarily bounded, with a smooth or empty boundary. If the parameter ε > 0 is small enough,the paper [22] proved the existence of nonconstant positive solutions under Neumann boundary conditions. Later, underthe Dirichlet conditions at the boundary, the paper [23] proved that (Pε) admits classical positive solutions uε, vε ∈

C2(Ω) ∩ C1(Ω) ∩ H10 (Ω) for any ε > 0 small enough. Moreover, uε, vε concentrate at a prescribed finite number of local

minimumpoints of V (x), possibly degenerate, as ε → 0. In the paper [24], the authors proved that the least energy solutionsto the system (Pε) concentrate, as ε → 0, at a point ofΩ whichmaximizes the distance to the boundary ofΩ . For thewholespace RN , the papers [25,13] considered the system

(Nε)

−ε2u + V (x)u = Rv(x, u, v) in RN ,

−ε2v + V (x)v = Ru(x, u, v) in RN ,u(x) → 0 and v(x) → 0 as |x| → ∞.

In the paper [13], assume that R(x, u, v) = h1(x)|v|q + h2(x)|u|p, where the exponents p, q > 1 are below the criticalhyperbola, that is, 1

p+1 +1

q+1 >N−2N (N ≥ 3). The authors obtained positive solutions for the system (Nε). Later, assume

that R(x, u, v) = K(x) v0 g(s)ds+H(x)

u0 f (s)ds and f , g have superlinear and subcritical growth at infinity, and V , H , K are

positive and locally Hölder continuous. Ramos [25] proved the existence and multiplicity positive solutions of the system(Nε). Nearly, the paper [26] considered nonperiodic superquadratic case for the system (Pε). By applying the generalized

342 L. Xiao et al. / J. Math. Anal. Appl. 399 (2013) 340–351

linking theorem of [18], the authors proved that the system (Pε) has existence of multiple solutions for ε > 0 small enough.As far as we know, for nonperiodic asymptotically linear case, there is still no result. The main purpose of this paper is toconsider such case. Themaindifficulty of this problem is the lack of compactness of Sobolev’s embedding theorem.Moreover,the variational functional is strongly indefinite. In such case, it is difficult for us to get the compactness for the Cerami-sequences. In order to overcome this difficulty, we manage to recover sufficient compactness by imposing a control on thesize of H(x, z) with respect to the behavior of V (x) at infinity in x (see the following condition (R3)). This approach wasoriginally developed by Jeanjean [27] (see also [28]).

Set Aλ := −∆+ λV (x). Denote by σ(A), σd(Aλ) and σe(Aλ) the spectrum, the point spectrum and the essential spectrumof the operator Aλ, respectively. Suppose that V and H satisfy the following conditions:

(V0) V (x) ∈ C(RN , [0,∞)), and there exists b > 0 such that Hb := x ∈ RN: V (x) < b is nonempty and has finite

measure;(R0) H ∈ C1(RN

× R2,R+) and Hz(x, z) = o(|z|) as |z| → 0 uniformly in x;(R1) H(x, z) =

12F(x)z · z + R(x, z), where F(x) ∈ L∞(RN ,R) and Rz (x,z)

|z| → 0 as |z| → ∞;(R2) F0 := infx∈RN F(x) > λ := inf[σ(Aλ) ∩ (0,+∞)];(R3) H∗

:= lim sup|x|→∞ supz=0|Hz (x,z)|

|z| < b0, where b0 := supb : |Hb| < ∞;(R4) One of the following holds:(i) 0 ∈ σ(J0Aλ − λF(x)) for λ > 0;(ii) H(x, z) ≥ 0 for all (x, z) and H(x, z) ≥ α0 for some α0 > 0 and for all (x, z) with |z| large enough, where

H(x, z) :=12Hz(x, z)z − H(x, z).

Our main result is the following.

Theorem 1.1. Let (V0), (R0) –(R4) be satisfied. Then there existsΛ0 > 0 such that (Pλ) has at least one nontrivial solution foreach λ ∈ (Λ0,+∞). Moreover, if H is odd in z, then for each m ≥ 1, there existsΛm > 0 such that (Pλ) has at least m pairs ofnontrivial solutions for each λ ∈ (Λm,+∞).

Since λ = ε−2, then by Theorem 1.1, we know that the following results hold.

Theorem 1.2. Let (V0), (R0) –(R4) be satisfied. Then there exists E0 > 0 such that (Pε) has at least one nontrivial solution foreach ε ∈ (0, E0). Moreover, if H is odd in z, then for each m ≥ 1, there exists Em > 0 such that (Pε) has at least m pairs ofnontrivial solutions for each ε ∈ (0, Em).

Remark 1.3. The following function satisfies (R0)–(R1) and (R4);Ex1. H(x, z) := b(x)|z|2(1 −

1ln2(e+|z|)

), where 0 < inf b(x) < sup b(x) < ∞ and b(x) ∈ C(RN ,R).

Notations. Form now on, the letter c will be indiscriminately used to denote various constants whose exact values areirrelevant.

2. Variational setting

In this section, we shall establish variational framework for the system (Pε). Let X and Y denote two Banach spaceswith norms ∥ · ∥X and ∥ · ∥Y , we always choose the equivalent norm ∥(x, y)∥X×Y := (∥x∥2

X + ∥y∥2Y )

12 on the product space

X × Y . In particular, if X and Y are two Hilbert spaces with inner products (·, ·)X and (·, ·)Y , we choose the inner product((x, y), (w, z))X×Y = (x, w)X + (y, z)Y . For the convenience of notation, let | · |q denote the usual Lq-norm and (·, ·)2 be theusual H := L2(RN)-inner product. In order to prove Theorem 1.1, we introduce the space

E+ :=

u ∈ H1(RN) :

RN

V (x)u2 < ∞

,

which is a Hilbert space equipped with the inner product

(u1, u2)+ :=

RN(∇u1∇u2 + V (x)u1u2)

and the associated norm ∥u∥2+

= (u, u)+. By the assumption (V0), it is easy to check that E+ embeds continuously inH1(RN).Note that the norm ∥ · ∥+ is equivalent to the one ∥ · ∥λ+ deduced by the inner product

(u1, u2)λ+ =

RN(∇u1∇u2 + λV (x)u1u2)

for each λ > 0. It is clear that, for each s ∈ [2, 2∗], there is ℓs > 0 (independent of λ) such that if λ ≥ 1

|u|s ≤ ℓs∥u∥+ ≤ ℓs∥u∥λ+ for all u ∈ E+.

Let λe := inf σe(A). Then we have the following lemma.

L. Xiao et al. / J. Math. Anal. Appl. 399 (2013) 340–351 343

Lemma 2.1. Suppose (V0) holds. Then λe ≥ λb0.

Proof. We follow the ideal of the proof in [29,30]. Let b > 0 such that |Hb| < ∞. Set Wλ(x) = λ(V (x) − b), W±

λ :=

max±Wλ(x), 0 and Aλ := − +λb + W+

λ . It follows from (V0) that the multiplicity operatorW−

λ is compact relative toAλ (cf. [31]). By Weyl’s theorem, we know that

σe(Aλ) ⊂ σe(Aλ) ⊂ [λb,+∞).

Since the claim is true for any b > 0 with |Hb| < ∞, one sees that σe(Aλ) ⊂ [λb0,+∞), as required.

Remark 2.2. Note that it is possible that λe = ∞, i.e., σd(Aλ) = σ(Aλ). For example, this is the case if V (x) → ∞ as|x| → ∞.

From now on, we fix a number bwith

H∗ < b < b0, (2.1)

where H∗ is defined in (R3). Let kλ be the number of eigenvalues less than λb. We write ηλi and hλi (1 ≤ i ≤ kλ) for theeigenvalues and eigenfunctions, respectively. Setting

Hdλ := spanhλ1, . . . , hλkλ,

we have the orthogonal decomposition

H := Hdλ ⊕ H e

λ, u = ud+ ue.

Correspondingly, E+ has the decomposition:

E+ := Edλ+ ⊕ Ee

λ+ with Edλ+ = Hd

λ and Eeλ+ = E+ ∩ H e

λ,

orthogonal with respect to both inner products (·, ·)2 and (·, ·)λ+. By Lemma 2.1, we have that

λb0|u|22 ≤ ∥u∥2λ+, ∀u ∈ Ee

λ+. (2.2)

Let S be the best Sobolev constant such that S|u|22∗ ≤

RN |∇u|2. It is clear that

S|u|22∗ ≤ ∥u∥2λ+, ∀u ∈ E+.

This together with interpolation implies that for each s ∈ [2, 2∗]

|u|ss ≤ asλ−

2∗−s2∗−2 ∥u∥s

λ+ for all u ∈ Eeλ+ (2.3)

where as is a constant independent of λ.In the sequel, we write |z| = (|u|2 + |v|2)

12 for z = (u, v). Let

E := E+⊕ E−,

where

E+:= E+ × 0 and E−

:= 0 × E+.

For z = (u, v) ∈ E, we write z+:= (u, 0) ∈ E+ and z−

:= (0, v) ∈ E−. Obviously, E is a Hilbert space under the innerproduct

(z1, z2) := (u1, u2)+ + (v1, v2)+

and the induced norm

∥z∥2:= ∥u∥2

++ ∥v∥2

+.

Clearly, E is a Hilbert space with the equivalent inner product

(z1, z2)λ := (u1, u2)λ+ + (v1, v2)λ+

and the norm

∥z∥2λ := ∥u∥2

λ+ + ∥v∥2λ+.

Set Eeλ := Ee

λ+ × Eeλ+ and Ed

λ := Edλ+ × Ed

λ+. Then E also has the following orthogonal decomposition

E = Eeλ ⊕ Ed

λ.

Accordingly, we write z = ze + zd for z ∈ E with zd = (ud, vd) and ze = (ue, ve). Note that dim Edλ < +∞. It follows

from (2.2) and (2.3) that for each s ∈ [2, 2∗]

|z|22 ≤1

b0λ∥z∥2

λ and |z|ss ≤ asλ−

2∗−s2∗−2 ∥u∥s

λ for all z ∈ Eeλ, (2.4)

where as is a constant independent of λ.

344 L. Xiao et al. / J. Math. Anal. Appl. 399 (2013) 340–351

Lemma 2.3. Assume (V0) is satisfied. Then E embeds continuously into Lp(RN ,R2) (∀p ∈ [2, 2∗]). Moreover, the embedding

E → Lploc(RN ,R2) (∀p ∈ [2, 2∗)) is compact.

Proof. It follows from E+ embeds continuously in H1(RN) that

c|u|22 ≤ ∥u∥2+, ∀u ∈ E+.

Moreover, we have

S|u|2∗

2∗ ≤ ∥u∥2∗

+, ∀u ∈ E+,

where S be the best Sobolev constant such that S|u|22∗ ≤

RN |∇u|2. This together with interpolation implies that for eachs ∈ [2, 2∗

]

c|u|ss ≤ ∥u∥s+, ∀u ∈ E+. (2.5)

Since |z| = (|u|2 + |v|2)12 for z ∈ E, then by (2.5), it follows that for all s ∈ [2, 2∗

]

c|z|2s ≤ ∥z∥2, ∀z ∈ E. (2.6)

Thus, we have that E embeds continuously into Lp(RN ,R2) (∀p ∈ [2, 2∗]). Moreover, since the embedding H1(RN) →

Lploc(RN) (∀p ∈ [2, 2∗)) is compact, it follows that the embedding E → Lploc(R

N ,R2) (∀p ∈ [2, 2∗)) is compact.

Obviously, the system (Pλ) has the following energy functional

Φλ(z) = Φλ(u, v) =

RN

12(|∇u|2 + λV (x)u2

− |∇v|2 − λV (x)v2)− λΨ (z)

for all z = (u, v) ∈ E, where Ψ (z) =

RN H(x, z). From the above space decomposition, it follows that the functional Φλcan be rewritten in a standard way

Φλ(z) = Φλ(u, v) =12∥z+

∥2λ −

12∥z−

∥2λ − λΨ (z) =

12∥u∥2

λ+ −12∥v∥2

λ+ − λΨ (z),

for all z = (u, v) = z++ z−

∈ E. By assumptions and Lemma 2.3, we know thatΨ ∈ C1(E,R) andΦ ∈ C1(E,R). Moreover,a standard argument shows that the critical points ofΦλ are solutions of (Pλ) (see e.g., [29,7,4]).

3. The abstract critical point theorem

In order to study the critical points ofΦ , we now recall some abstract critical point theory developed recently in [29,18].Also see [19,20] for earlier related results.

Let (E, ∥ · ∥) be a Banach space with direct sum decomposition E = X ⊕ Y , and PX , PY denote the projections onto X ,Y , respectively. For a functional Φ ∈ C1(E,R), we write Φa := z ∈ E : Φ(z) ≥ a, Φb

:= z ∈ E : Φ(z) ≤ b andΦb

a = Φa ∩ Φb. Next, let us recall some definitions:(i)Φ is said to be weakly sequentially upper semi-continuous, i.e., if zn z in E one hasΦ(z) ≥ lim infn→∞Φ(zn);(ii)Φ ′ is said to be weakly sequentially continuous, i.e., if zn z, limn→∞Φ ′(zn)w = Φ ′(z)w for eachw ∈ E;(iii) a sequence zn ⊂ E is said to be a (C)c-sequence if Φ(zn) → c and (1 + ∥zn∥)Φ ′(zn) → 0; Φ is said to satisfy the

(C)c-condition if any (C)c-sequence has a convergent subsequence.From now on we assume that the Banach space X is separable and reflexive, and fix a countable dense subset B ⊂ X∗.

For each b ∈ B we define a semi-norm on E by

Pb : E = X ⊕ Y → R, Pb(x + y) = qb(x)+ ∥y∥, for x + y ∈ X ⊕ Y ,

where qb(x) = |(x, b)X,X∗ | = |b(x)|. We denote by TB the induced topology. Letw∗ denote the weak∗-topology on E∗.Assume the following.(K0) For any c > 0, there exists ξ > 0 such that ∥z∥ < ξ∥PY z∥ for all z ∈ Φc .(K1) For any c ∈ R,Φc is TB-closed, andΦ ′

: (Φc, TB) → (E∗, w∗) is continuous.(K2) There exists ϱ > 0 with κ := infΦ(SϱY ) > 0 where SϱY := z ∈ Y : ∥z∥ = ϱ.Then we have the following theorem.

Theorem 3.1. Let (K0) –(K2) be satisfied and suppose there are R > ϱ > 0 and e ∈ Y with ∥e∥ = 1 such that supΦ|∂M ≤ κwhere M := z = x + se; s ≥ 0, x ∈ X, ∥z∥ ≤ R. Then Φ has a (C)c-sequence with κ ≤ Φ(z) ≤ c := supΦ|M . Moreover, ifΦ satisfies the (C)c-condition for all c ≤ c , thenΦ has a critical point z with κ ≤ Φ(z) ≤ c .

L. Xiao et al. / J. Math. Anal. Appl. 399 (2013) 340–351 345

This theorem is a special case of Theorem 4.4 in [32]. For stating the multiple critical points theorem for the functionalΦ , we further assume the following.(K3) There are a finite-dimensional subspace Y0 ⊂ Y and R > ϱ such that for E0 := X ⊕ Y0 and B0 := u ∈ E0; ∥u∥ ≤ R,

we have c := supΦ|E0 < ∞ and supΦ|E0\B0 < infΦ|Bϱ∩Y , where Bϱ = z ∈ E : ∥z∥ ≤ ϱ.The following theorem is a special case of Theorem 4.6 in [18].

Theorem 3.2. If Φ is even, and satisfies (K1) –(K3) and the (C)c-condition for all c ∈ [κ, c], then it has at least M := dimY0pairs of critical points with critical values less than or equal to c.

4. The linking structure

In this section we study the linking structure for the functionalΦλ. Clearly, under the assumptions (R0)–(R2), we knowthat for any ϵ > 0, there is Cϵ > 0 such that

Hz(x, z) ≤ ϵ|z| + Cϵ |z|p−1 (4.1)and

H(x, z) ≤ ϵ|z|2 + Cϵ |z|p (4.2)for (x, z), where p ∈ [2, 2∗). First we have the following lemma.

Lemma 4.1. Suppose that the conditions (V0) and (R0) –(R2) are satisfied. Then for each λ ≥ 1, there exists ϱλ > 0 such thatκλ := infΦλ|∂Bϱλ∩E+ > 0.

Proof. For z+= (u, 0) ∈ E+, by (4.2) and Lemma 2.3 we have

Φλ(z+) =12∥z+

∥2− λ

RN

H(x, z+) ≥12∥z+

∥2− λϵ|z+

|22 − λCϵ |z+

|pp

≥

12

− cλϵ

∥z+∥2− λcCε∥z+

∥p.

Let ϵ < 14λc , we know that the lemma holds with ∥z+

∥λ small enough.

Lemma 4.2. Assume that (V0) and (R0) –(R2) hold. Then for each m ∈ N, there exists Nm > 0 such that Aλ has at least meigenvalues (counted with multiplicity) lying in (0, λF0) for all λ ≥ Nm, where F0 is given in (R2).

Proof. Similar to Lemma 5.2 of [33], we will establish this lemma constructively. By Lemma 2.1, we know that σe(Aλ) ⊂

R \ (−∞, λb0]. Thus it is sufficient to show that there are m linearly independent elements ϕ ∈ E+ with |ϕ|2 = 1 and∥ϕ∥λ+ < λF0 < λb0. Set 0 < θ < 1,

Gθ :=

x ∈ RN

:θF02

≤ V (x) ≤ θF0

and Ωθ := int Gθ .

For eachm ∈ N, we choosem real functions ωj ∈ C∞

0 (Ωθ ,R), j = 1, 2, . . . ,m, satisfying

|ωj|2 = 1 and supp ωj ∩ supp ωk = ∅ if j = k.

It is clear that ω1, . . . , ωm are linearly independent. Moreover,

|∇ωj|22 +

λF0θ2

≤ (Aλωj, ωj)2 = |∇ωj|22 + λ

RN

V (x)|ωj|2

≤ |∇ωj|22 + λF0θ. (4.3)

Set

Nm := spanω1, . . . , ωm.

Obviously, one sees that dim Nm = m. In the following, we set

δ = max|∇ωj|22 : j = 1, 2, . . . ,m.

Clearly, δ depends on m and the choice of ωj, but is independent of λ. Denote u =m

j=1 cjωj ∈ Nm, it is clear that|u|22 =

mj=1 c

2j . Similar to (4.3), we have that

λF0θ2

|u|22 ≤ ∥u∥2λ+ = (Aλu, u)2 =

mj=1

c2j (Aλωj, ωj)2

≤ (δ + λF0θ)|u|22 = [λF0 + δ + λF0(θ − 1)]|u|22,

346 L. Xiao et al. / J. Math. Anal. Appl. 399 (2013) 340–351

and

∥u∥2λ+ − λF0|u|22 ≤ [δ + λF0(θ − 1)]|u|22.

Since 0 < θ < 1 and F0 > 0, then there is Nm > 0 such that

∥u∥2λ+ − λF0|u|22 ≤ −γθ |u|22, ∀λ ∈ [Nm,∞),

where γθ > 0. From the above, we obtain that for all λ ≥ Nm

µm(Aλ|L2) := inf(F⊂E+,dim(F)=m)

sup(η∈F ,|η|2=1)

(Aλη, η)2

≤ supu∈Nm,|u|2=1

(Aλu, u)2

≤ supu∈Nm,|u|2=1

(λF0 − γθ )

< λF0.

The conclusion follows.

Next, we study the linking structure of the function Φλ. For each m ≥ 1, by Lemma 4.2, there exists Nm > 0 suchthat, for each λ > Nm, Aλ has at least m eigenvalues (counted with multiplicity) lying in (0, λF0). We arrange themas 0 < µ1 ≤ · · · ≤ µm < λF0. Let ej denote the corresponding eigenfunctions: Aλej = µjej for j = 1, . . . ,m. SetW0 := span(e1, 0), . . . , (em, 0). Obviously,

µ1|u|22 ≤ ∥u∥2λ ≤ µm|u|22 < λF0|u|22 for all u ∈ W0. (4.4)

For any finite-dimensional subspace W of W0, set EW = E−⊕ W .

Lemma 4.3. Under the assumptions of Lemma 4.1, we have that for each λ ≥ Nm, and any subspaceW of W0, supΦλ|EW < ∞.Moreover, there exists RW (λ) > 0 such that Φλ(z) < infΦλ|Bϱλ∩E+ for all z ∈ EW with ∥z∥ ≥ RW (λ), where ϱλ > 0 is givenin Lemma 4.1.

Proof. It suffices to prove that Φλ(z) → −∞ in EW as ∥z∥λ → ∞. If not, then there are Q > 0 and zj ⊂ EW with∥zj∥λ → ∞ such that Φλ(zj) ≥ −Q for all j, where zj = (uj, vj). Denote yj :=

zj∥zj∥λ

, passing to a subsequence if necessary,

yj y, y−

j y− and y+

j → y+. Since H(x, z) ≥ 0, then we have

12(∥y+

j ∥2λ − ∥y−

j ∥2λ) ≥

12(∥y+

j ∥2λ − ∥y−

j ∥2λ)− λ

RN

H(x, zj)∥zj∥2

λ

=Φλ(zj)∥zj∥2

λ

≥−Q∥zj∥2

λ

, (4.5)

which yields that

12∥y−

j ∥2λ ≤

12∥y+

j ∥2λ +

Q∥zj∥2

λ

. (4.6)

We claim that y+= 0. Indeed, if not, it follows from (4.6) that ∥y−

j ∥ → 0. Thus ∥yj∥ → 0, which contradicts with ∥yj∥ = 1.By (4.4) one has

∥y+∥2λ − ∥y−

∥2λ − λ

RN

F(x)|y|2 ≤ ∥y+∥2λ − ∥y−

∥2λ − λF0|y|22

≤ −(λF0 − µm)|y+|22 < 0,

then there exists R > 0 such that

∥y+∥2λ − ∥y−

∥2λ − λ

BR

F(x)|y|2 < 0, (4.7)

where BR := x ∈ RN: |x| ≤ R. It follows from (R1) that |R(x, z)| ≤ c|z|2 for some c > 0, and R(x,z)

|z|2→ 0 as |z| → ∞

uniformly in x. Note that

limj→∞

BR

R(x, zj)∥zj∥2

λ

= limj→∞

BR

R(x, zj)|zj|2

|yj|2 = 0. (4.8)

L. Xiao et al. / J. Math. Anal. Appl. 399 (2013) 340–351 347

Thus Eqs. (4.7)–(4.8) imply that

0 ≤ limj→∞

12(∥y+

j ∥2λ − ∥y−

j ∥2λ)− λ

BR

H(x, zj)∥zj∥2

λ

≤

12

∥y+

∥2λ − ∥y−

∥2λ − λ

BR

F(x)|y|2

< 0.

Now the desired conclusion follows from this contradiction.

Especially, we have the following lemma.

Lemma 4.4. Let the assumptions of Lemma 4.1 be satisfied. Then letting (e, 0) ∈ W0 with ∥(e, 0)∥λ = 1, there is Rλ > 0 andN0 > 0 such that Φλ|∂M ≤ κλ for all λ ≥ N0, where M := z = z−

+ se : z−∈ E−, s ≥ 0, ∥z∥λ ≤ Rλ, and κλ is defined

in Lemma 4.1.

5. The (C)c-sequence

In this section we shall prove thatΦλ satisfies the (C)cλ-condition.

Lemma 5.1. Let (V0) and (R0) –(R4) be satisfied. Then for each λ ≥ 1, we have that any (C)cλ-sequence is bounded.Proof. Let zj ⊂ E be such that

Φλ(zj) → cλ and (1 + ∥zj∥λ)Φ ′

λ(zj) → 0. (5.1)

Then, for C0(λ) > 0, one has

C0(λ) ≥ Φλ(zj)−12Φ ′

λ(zj)zj = λ

RN

H(x, zj). (5.2)

To prove that zj is bounded, we develop a contradiction argument related to the one introduced in [34] (see also [28,35]).We assume that, up to a subsequence, ∥zj∥λ → ∞ as j → ∞. Set yj :=

zj∥zj∥λ

. Then ∥yj∥λ = 1. Without loss of generality, we

can assume that yj y := (ξ , ζ ) in E, yj → y in Lploc for p ∈ [2, 2∗), and yj(x) → y(x) for a.e. x ∈ RN . There are only twocases needed to be considered: y ≡ 0 or y = 0.

If y ≡ 0, then yj 0 in E and yj → 0 in Lploc(RN ,R2) for p ∈ [2, 2∗). Moreover, ydj → 0 in E and ydj → 0 in Lploc .

Choose some number b such that H∗ < b < b0. It follows from (R3) that for fixed ϵ > 0, there exists Rϵ > 0 such that|Hz(x, z)| ≤ (b − ϵ)|z| for |z| ≥ Rϵ uniformly in x. Set Bϵ,j = x ∈ RN

: |zj(x)| ≤ Rϵ and Bcϵ,j = RN

\ Bϵ,j. It follows from

o(1) =Φ ′

λ(zj)(ze+j − ze−j )

∥zj∥2λ

= ∥yej ∥2λ − λ

RN

Hz(x, zj)|zj|

(ye+j − ye−j )|yj|

that

∥yej ∥2λ = λ

Bϵ,j

Hz(x, zj)|zj|

(ye+j − ye−j )|yj| + λ

Bcϵ,j

Hz(x, zj)|zj|

(ye+j − ye−j )|yj| + o(1)

≤ cλBϵ,j

|ye+j − ye−j ∥yj| + λ(b − ϵ)

Bcϵ,j

|ye+j − ye−j ∥yj| + o(1)

≤ λ(b − ϵ)|yej |22 + o(1). (5.3)

By (2.4) one gets1 −

b − ϵ

b

∥yej ∥

2≤ o(1),

which implies that ∥yej ∥2λ → 0. Hence 1 = ∥yj∥2

λ = ∥yej ∥2λ + ∥ydj ∥

2λ → 0, a contradiction. Therefore, y ≡ 0 cannot occur.

Now suppose y = (ξ , ζ ) = 0. For each ϕ = (ψ, φ) ∈ C∞

0 (RN ,R2), we have

Φ ′

λ(zj)ϕ = (z+

j − z−

j , ϕ)λ − λ

RN

Hz(x, zj)ϕ

= (z+

j − z−

j , ϕ)λ − λ

RN

Rz(x, zj)ϕ − λ

RN

F(x)zjϕ

= ∥zj∥λ

(y+

j − y−

j , ϕ)− λ

RN

Rz(x, zj)ϕ|zj|

|yj| − λ

RN

F(x)yjϕ,

348 L. Xiao et al. / J. Math. Anal. Appl. 399 (2013) 340–351

which yields that

o(1) = (y+

j − y−

j , ϕ)λ − λ

RN

Rz(x, zj)ϕ|zj|

|yj| − λ

RN

F(x)yjϕ.

Letting j → ∞, we have (y+− y−, ϕ)λ − λ(F(x)y, ϕ)2 = 0, i.e.,

(Fλ)

−ξ + λV (x)ξ = λF(x)ξ in RN ,

−ζ + λV (x)ζ = −λF(x)ζ in RN .

Recall that J0 =

1 00 −1

. Thenthe system (Fλ) can be rewritten as

−∆y + λV (x)y = J0λF(x)y.

This is impossible if (i) of (R4) is satisfied. Now we assume (ii) of (R4) holds. We adopt an argument borrowing from Dingand Jeanjean [28]. Let ω :=

ξ+ζ

2 and ψ :=ξ−ζ

2 . Then the system (Fλ) can be written as

(Iλ)

−ω + λV (x)ω = λF(x)ψ in RN ,

−ψ + λV (x)ψ = λF(x)ω in RN .

Since y = (ξ , ζ ) = 0, then χ = (ω,ψ) = 0. Thus, either ω = ψ or ω ≡ ψ = 0 holds.(I) If ω = ψ , by (Iλ) one has

−∆φ + λV (x)φ = −λF(x)φ, for φ := ω − ψ = 0. (5.4)

(II) If ω ≡ ψ = 0, it follows from (Iλ) that

−∆ω + λV (x)ω = λF(x)ω, for ω = ψ = 0. (5.5)

From the condition (ii) of (R4), there exists some α > 0 such that H(x, z) ≥ α0 whenever |z| ≥ α. Clearly, one has

C0(λ) ≥ Φλ(zj)−12Φ ′

λ(zj)zj = λ

RN

H(x, zj)

≥ λ

|zj|≥α

H(x, zj) ≥ λ

|zj|≥α

α0 = λα0|x ∈ RN: |zj| ≥ α|;

hence

|x ∈ RN: |zj(x)| ≥ α| ≤

C0(λ)

λα0.

(i) If (I) holds, set Ξ := x ∈ RN: φ(x) = ω(x)− ψ(x) = 0. By (5.4) and the unique continuation arguments similar to

those of [36–39], we deduce that |Ξ | = ∞. Hence there exists ε > 0 andΞ ′⊂ RN such that |y(x)| = |(ω+ψ,ω−ψ)| ≥ 2ε

for x ∈ Ξ ′ and 2C0(λ)λα0

≤ |Ξ ′| < ∞. By Egoroff’s theorem, we can find a set Ξ ′′

⊂ Ξ ′ with |Ξ ′′| >

C0(λ)λα0

such that yj → yuniformly onΞ ′′. So for almost all j, |yj(x)| ≥ ε and |zj| ≥ α inΞ ′′. Then

C0(λ)

λα0< |Ξ ′′

| ≤ |x ∈ RN: |zj(x)| ≥ α| ≤

C0(λ)

λα0. (5.6)

A contradiction.(ii) If (II) holds, let Ξ := x ∈ RN

: ω(x) = ψ(x) = 0. Repeating the argument of (i), we can acquire the contradictioninequality (5.6). Thus, the sequence zj is bounded.

Let zj ⊂ E be an arbitrary (C)cλ-sequence. By Lemma5.1 it is bounded; hence,wemay assumewithout loss of generalitythat zj z in E, zj → z in Lqloc for q ∈ [2, 2∗) and zj(x) → z(x) a.e. on RN . Obviously, z is a critical point ofΦλ.

Lemma 5.2. Let s ∈ [2, 2∗). There is a subsequence such that for any ϵ > 0, there exists hϵ > 0 such that

lim supn→∞

Bn\Br

|zjn |s≤ ϵ (5.7)

for all r ≥ hϵ , where BR := x ∈ RN; |x| ≤ R.

L. Xiao et al. / J. Math. Anal. Appl. 399 (2013) 340–351 349

Proof. We follow the ideal of [28]. Note that, for each n ∈ N,Bn

|zj|s →Bn

|z|s as j → ∞, there exists jn ∈ N such thatBn(|zj|s − |z|s)dt <

1n

for all j = jn + i, i = 1, 2, . . . .

Without loss of generality we can assume jn+1 ≥ jn. In particular, for jn = jn + n we haveBn(|zjn |

s− |z|s) <

1n.

Clearly, there exists hϵ satisfyingRN\Br

|z|s < ϵ, ∀r ≥ hϵ . (5.8)

Since Bn\Br

|zjn |s=

Bn(|zjn |

s− |z|s)+

Bn\Br

|z|s +

Br(|z|s − |zjn |

s)

≤1n

+

RN\Br

|z|s +

Br(|z|s − |zjn |

s),

by (5.8), the lemma now follows easily.

In the following, we adopt a cut-off technique developed by Ackermann [32,40]. Let η : [0,∞] → [0, 1] be a smoothfunction satisfying η(t) = 1 if t ≤ 1, η(t) = 0 if t ≥ 2. Define zj(x) = η(

2|x|j )z(x). Clearly,

∥zj − z∥λ → 0, as j → ∞. (5.9)

In a similar way to Lemma 3.2 in Ackermann [32] (see also Lemma 4.4 in [28]), we can obtain the following.

Lemma 5.3. Under the assumptions of Theorem 1.1, we have

sup∥ϕ∥≤1

RN(Hz(x, zjn)− Hz(x, zjn − zn)− Hz(x, zn))ϕ

→ 0, as n → ∞.

Proof. Note that (5.9) and the local compactness of Sobolev’s embedding imply that, for any r > 0,

limn→∞

Br(Hz(x, zjn)− Hz(x, zjn − zn)− Hz(x, zn))ϕ

= 0

uniformly in ∥ϕ∥ ≤ 1. For any ϵ > 0, we infer from (5.8) and (5.9) that

limn→∞

Bn\Br

|zjn |s≤

RN\Br

|z|s ≤ ϵ, for all r ≥ hϵ .

Using (5.7) for s = 2, pwe have

lim supn→∞

RN(Hz(x, zjn)− Hz(x, zjn − zn)− Hz(x, zn))ϕ

= lim sup

n→∞

Bn\Br

(Hz(x, zjn)− Hz(x, zjn − zn)− Hz(x, zn))ϕ

≤ C1 lim supn→∞

Bn\Br

(|zjn | + |zn|)|ϕ| + C2 lim supn→∞

Bn\Br

(|zjn |p−1

+ |zn|p−1)|ϕ|

≤ C1 lim supn→∞

(|zjn |L2(Bn\Br ) + |zn|L2(Bn\Br ))|ϕ|2 + C2 lim supn→∞

(|zjn |p−1Lp(Bn\Br ) + |zn|

p−1Lp(Bn\Br ))|ϕ|p

≤ C3ϵ12 + C4ϵ

p−1p

uniformly in ∥ϕ∥ ≤ 1, and this proves the lemma.

Lemma 5.4. Let (V0), (H0) –(H4) be satisfied. Then for each λ ≥ 1,Φλ satisfies the (C)cλ-condition.

350 L. Xiao et al. / J. Math. Anal. Appl. 399 (2013) 340–351

Proof. Let zj be a (C)cλ-sequence of Φλ. Now we use the decomposition E = Eeλ ⊕ Ed

λ . Recall that dim Edλ < ∞. Set

yn := zjn − zn = yen + ydn. Then yn 0 and ydn = (zdjn − zd)+ (zd − zdn) → 0. Moreover, it follows that for ϕ ∈ E

Φ ′

λ(yn)ϕ = Φ ′

λ(zjn − zn)ϕ = Φ ′

λ(zjn)ϕ − Φ ′

λ(zn)ϕ + λ

RN(Hz(x, zjn)− Hz(x, zjn − zn)− Hz(x, zn))ϕ.

From Lemma 5.3 it follows that, letting n → ∞, one hasΦ ′

λ(yn) → 0. Set yen := ye+n − ye−n . Observe that

o(1) = Φ ′

λ(yn)yen = ∥yen∥

2− λ

RN

Hz(x, yn)yen.

Thus, similar to (5.3), we have

∥yej ∥2

= λ

Bϵ,j

Hz(x, yj)|yj|

(yej )|yj| + λ

Bcϵ,j

Hz(x, yj)|yj|

(yej )|yj| + o(1)

≤ λcBϵ,j

|yej ∥yj| + λ(b − ϵ)

Bcϵ,j

|yej ∥yj| + o(1)

≤ λ(b − ϵ)|yej |22 + o(1) ≤

b − ϵ

b∥yej ∥

2λ + o(1).

Hence ∥yej ∥2λ → 0, and so ∥yn∥λ → 0. Since zjn − z = yn + (zn − z), hence ∥zjn − z∥λ → 0. This ends the proof.

6. Proof of Theorem 1.1

In order to apply the abstract Theorems 3.1–3.2 to Φλ, we choose X = E− and Y = E+. X is separable and reflexive andlet B be a countable dense subset of X∗. First we have the following lemma.

Lemma 6.1. Φλ satisfies (K0).

Proof. For any c > 0, z ∈ Φλc , i.e.,Φλ(z) ≥ c , it follows from H ≥ 0 that

0 < c ≤12(∥z+

∥2λ − ∥z−

∥2λ).

This yields ∥z−∥λ < ∥z+

∥λ, and hence ∥z∥λ < 2∥z+∥λ.

Lemma 6.2. Φλ satisfies (K1) for all λ ≥ 1.

Proof. We first show that Φλa is TB-closed for every a ∈ R. Consider a sequence zn ⊂ Φλa which TB-converges to z ∈ E,and write zn = z+

n + z−n , z = z−

+ z+. Then z+n → z+ is in norm topology and hence z+

n is bounded in norm topology.Observe that there exists C(λ) > 0 such that

C(λ) ≥ ∥z+

n ∥2λ − 2Φλ(zn)− 2λ

RN

H(x, zn) = ∥z−

n ∥2λ.

This implies the boundedness of z−n , and hence z−

n z−. Therefore, we have zn z. SinceΨ is weakly sequentially lowersemi-continuous, it follows that

a ≤ limnΦλ(zn) ≤ Φλ(z),

thus z ∈ Φλa andΦλa is TB-closed.Next we show thatΦ ′

λ : (Φλc, TB) → (E∗, w∗) is continuous. It suffices to show that Ψ ′ has the same property. Supposezn z in E. Then zn → z in Lploc for p ∈ [2, 2∗). It is obvious that

Ψ ′(zn)ϕ =

RN

Hz(x, zn)ϕ →

RN

Hz(x, z)ϕ = Ψ ′(z)ϕ

for all ϕ ∈ C∞

0 (RN), as n → ∞. Now using the density of C∞

0 (RN) in E we can obtain the desired conclusion.

Proof of Theorem 1.1. Existence: By Lemmas 6.1–6.2, we know that the conditions (K0)–(K1) are satisfied for all λ ≥ 1.Lemma 4.1 implies (K2) for all λ ≥ 1. Lemma 4.4 shows that Φλ possesses the linking structure for all λ ≥ N0, andLemma 5.4 implies that Φλ satisfies the (C)cλ-condition for all λ ≥ 1. Let Λ0 = max1,N0. Thus, we know that theconditions of Theorem 3.1 are satisfied for all λ ≥ Λ0. Therefore, by Theorem 3.1, we have that Φλ has at least one criticalpoint z with 0 < κλ ≤ Φλ(z).

Multiplicity: SinceH(x, z) is even in z, we know thatΦλ is even. Lemma 4.3 says thatΦλ satisfies (K3) for all λ ≥ Nm withdim W0 = m. Set Λm = max1,Nm. It follows that the conditions of Theorem 3.2 are satisfied for all λ ≥ Λm. Therefore,Φλ has at leastm pairs of nontrivial critical points by Theorem 3.2.

L. Xiao et al. / J. Math. Anal. Appl. 399 (2013) 340–351 351

Acknowledgments

The authors would like to thank the referees for giving valuable comments and suggestions, which make us possibleto improve the paper. This work was supported by Natural Science Foundation of China (11201186), Natural ScienceFoundation of Jiangsu Province (BK20122 82), Jiangsu University foundation grant (11JDG117), China Postdoctoral ScienceFoundation (2012 M511199), National Natural Science Foundation of China (Nos: 71073070, and 11071038), and theDoctoral Program of Higher Specialized Research Fund of China (20103227110015).

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