Infinitely many solutions for asymptotically linear periodic hamiltonian elliptic systems

DOI:10.1051/cocv:2008064 www.esaim-cocv.org

INFINITELY MANY SOLUTIONS FOR ASYMPTOTICALLY LINEAR PERIODIC HAMILTONIAN ELLIPTIC SYSTEMS

Fukun Zhao

, Leiga Zhao

and Yanheng Ding

Abstract. This paper is concerned with the following periodic Hamiltonian elliptic system

⎧⎨

⎩

−Δϕ+V(x)ϕ=Gψ(x, ϕ, ψ) inR^N,

−Δψ+V(x)ψ=Gϕ(x, ϕ, ψ) inR^N, ϕ(x)→0 andψ(x)→0 as|x| → ∞.

Assuming the potential V is periodic and 0 lies in a gap of σ(−Δ +V), G(x, η) is periodic in x and asymptotically quadratic in η = (ϕ, ψ), existence and multiplicity of solutions are obtained via variational approach.

Mathematics Subject Classification.35J50, 35J55.

Received March 11, 2008. Revised July 6, 2008.

Published online October 21, 2008.

1. Introduction and main results

Consider the following Hamiltonian elliptic system

⎧⎨

⎩

−Δϕ+V(x)ϕ=G_ψ(x, ϕ, ψ) in R^N,

−Δψ+V(x)ψ=G_ϕ(x, ϕ, ψ) inR^N,

ϕ(x)→0 andψ(x)→0 as|x| → ∞, (ES)

whereN ≥1,ϕ, ψ:R^N →R,V ∈C(R^N,R) andG∈C¹(R^N ×R²,R).

For the case of a bounded domain, assumingV ≡0, there are a number of papers concerned with the systems like or similar to (ES). For example, see Benci and Rabinowitz [8], De Figueiredo and Ding [11], De Figueiredo and Felmer [12], Hulshof and Van de Vorst [17] and their references for superlinear systems and earlier works, see Kryszewski and Szulkin [18] and the references therein for asymptotically linear systems, see Pistoia and Ramos [22] and their references for a singularly perturbed problem. Recently, De Figueiredoet al.[14] treated the system withG(x, ϕ, ψ) =F(ϕ) +H(ψ), and a nontrivial solution was obtainedviaan Orlitz space approach.

Keywords and phrases. Hamiltonian elliptic system, variational methods, strongly indefinite functionals.

∗ Supported by NSFC(10561011), NSFY of Yunnan Province and the Foundation of Education Commission of Yunnan Province, P.R. China.

1 Department of Mathematics, Yunnan Normal University, Kunming 650092 Yunnan, P.R. China.fukunzhao@163.com

2 Institute of Mathematics, AMSS, CAS, Beijing 100080, P.R. China.

3 Department of Mathematics, Beijing University of Chemical technology, Beijing 100029, P.R. China.

Article published by EDP Sciences c EDP Sciences, SMAI 2008

There are several authors who considered the systems on the whole space R^N. But most of them focused on the case V ≡1, which is not only radial but also periodic. The main difficulty of such type of problems is the lack of the compactness of the Sobolev embedding. An usual way to overcome the difficulty is imposing a radial symmetry assumption on the nonlinearities and working on the radially symmetric function space, which possesses a compact embedding. By this means, De Figueiredo and Yang [13] obtained a positive radially symmetric solution which decays exponentially to 0 at infinity. Their results were generalized by Sirakov [25] in a different way. Later, Bartsch and De Figueiredo [6] proved that the system admits infinitely many radial as well as non-radial solutions ifGis even inz. Li and Yang [21] proved,viaa generalized linking theorem, that (ES) has a positive ground state solution forV ≡1 and an asymptotically quadratic nonlinearityG(x, ϕ, ψ) =F(ϕ)+

H(ψ), and based on this result they obtained a positive solution for G(x, ϕ, ψ) =_ϕ

0 f(x, t)dt+_ψ

0 (x, s)ds if f(x, ϕ) andg(x, ψ) have autonomous limits ¯f(ϕ) and ¯h(ψ) at infinity. Very recently, Ding and Lin [16] considered semiclassical problems for systems of Schr¨odinger equations with subcritical and critical nonlinearities.

Another usual way is avoiding the indefinite character of the original functional by using the dual variational method, see for instance ´Avila and Yang [4,5], Alveset al.[3], Yang [28] and the references therein.

In this paper, we consider a periodic asymptotically linear elliptic system with 0 lying in a gap ofσ(S), where S :=−Δ +V is the Schr¨odinger operator, andσ(S) denotes the spectrum of the operator S. We must face two kinds of indefiniteness: one comes from the system itself and the other comes from each equation in the system. As to our knowledge, there is no multiplicity result for the asymptotically linear systems. The purpose of this paper is to obtain the existence and multiplicity of solutions. Since there is no radial assumption, we have to work onH¹(R^N)×H¹(R^N). Thanks to the periodic assumption, we can prove a weak version of the Cerami condition similar to Coti-Zelati and Rabinowitz [9,10]. By establishing a proper variational framework, we can obtain the multiplicity resultsviathe critical point theory of strongly indefinite functional, which were developed recently by Bartsch and Ding [7].

More precisely, for the potential V, we assume

(V₀) V ∈C(R^N,R) is 1-periodic in eachx_i fori= 1, . . . , N.

It is well known that, under (V₀),S is semibounded from below and the spectrumσ(S) =σ_cont(S) is a union of closed intervals (see Reed and Simon [23]), whereσ_cont(S) denotes the continuous spectrum of the operatorS.

The relationship between 0 and σ(S) is important to our approach. So we assume in addition that (V₁) 0 lies in a gap ofσ(S).

By (V₁), there holds

Λ := sup[σ(S)¯ ∩(−∞,0)]<0<Λ := inf[σ(S)∩(0,∞)].

In what follows, we use the notationη=: (ϕ, ψ). For the nonlinearity, we assume (G₀) G∈C¹(R^N ×R²,[0,∞)) is 1-periodic in each x_i fori= 1, . . . , N; (G₁) G(x, η) =o(|η|²) as|η| →0.

(G₂) |G_η(x, η)−G_∞(x)η|/|η| →0 as|η| → ∞, whereG_∞∈C(R^N,R) is 1-periodic in eachx_ifori= 1, . . . , N; (G₃) G₀:= inf

x∈R^NG_∞(x)>Λ, where Λ := max Λ,−Λ¯

;

(G₄) ˆG(x, η)>0 ifη= 0, and ˆG(x, η)→ ∞as|η| → ∞, where ˆG(x, η) =¹₂G_η(x, η)η−G(x, η);

(G₄) there exists δ₀ ∈ (0,Λ₀) such that ˆG(x, η) ≥ δ₀ whenever |G_η(x, η)| ≥ (Λ₀−δ₀)|η|, where Λ₀ :=

min

−Λ,¯ Λ

;

(G₅) ˆG(x, η)≥0, and there existsδ₁>0 such that ˆG(x, η)>0 if 0<|η| ≤δ₁.

Remark 1.1. There are some functions which satisfy (G₀)–(G₅). For example,G(x, η) =a(x)|η|²(1−_ln(e+|η|)¹ ), where infa(x)>Λ and is 1-periodic in eachx_i fori= 1, . . . , N.

Observe that, due to the periodicity ofV, G, if (ϕ, ψ) is a solution of (ES), then so is (a∗ϕ, b∗ψ) for each a, b∈Z^N, where (a∗ϕ)(x) =ϕ(x+a) and (b∗ψ)(x) =ψ(x+b). Two solutions (ϕ₁, ψ₁) and (ϕ₂, ψ₂) are said to be geometrically distinct ifa∗ϕ₁=ϕ₂ andb∗ψ₁=ψ₂ for alla, b∈Z^N. Our main result is the following:

Theorem 1.1. Let (V₀)–(V₁), (G₀)–(G₃),(G₅) and(G₄)or (G₄) be satisfied. Then (ES) has a least energy solution. If additionally G(x, η)is even in η then (ES) has infinitely many geometrically distinct solutions.

The paper is organized as follows. In Section 2, we set up the framework in which we study the variational problem associated to (ES). The linking structure of the functional will be discussed in Section 3. Some properties of (C)_c sequences will be showed in Section 4. The proof of Theorem 1.1 announced above will be given in the last section.

2. Variational setting

Below by | · |_q we denote the usual L^q-norm, c or c_i stand for different positive constants. Let X and Y be two Banach spaces with norms · _X and · _Y, we always choose the equivalent norm (x, y)_X×Y = (x²_X+y²_Y)^1/2 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 on the product spaceX×Y.

Let{E_λ}_λ∈Rbe the spectral family ofS. Assumption (V₁) implies an orthogonal decomposition:

L²:=L²(R^N,R) =L⁺⊕L⁻, z=z⁻+z⁺,

whereL⁻=E₀L²andL⁺= (Id−E₀)L². Denoting by|S|the absolute value ofS and its square root operator is

|S|^1/2= _∞

−∞|λ|dE(λ) :D(|S|^1/2)→L², where

D(|S|^1/2) = u∈L²| _∞

−∞|λ|d(E(λ)u, u)_L² <∞

. LetH =D(|S|^1/2) be the Hilbert space with the inner product

(u, v)_H= (|S|^1/2u,|S|^1/2v)_L2

and the corresponding normuH= (u, u)^1/2_H . There is an induced decomposition H=H⁻⊕H⁺, H^±=H∩L^±,

which is orthogonal with respect to the inner products (·,·)_L2 and (·,·)_H. Then for anyu∈H,u=u⁺+u⁻, u^±∈H^±, there holds

R^N|∇u|²+V(x)u²=u⁺²_H− u⁻²_H. (2.1) LetE=H×H with the inner product

((u, v),(ϕ, φ)) = (u, ϕ)_H+ (v, ψ)_H and the corresponding norm

(u, v)= [u²_H+v²_H]^1/2.

Recall thatE →L^p(R^N,R²) is continuous forp∈[2,2^∗] andE →L^p_loc(R^N,R²) is compact forp∈[2,2^∗), where 2^∗ is the Sobolev critical exponent. OnE we define the following functional

I(η) =I(ϕ, ψ) =

R^N∇ϕ∇ψ+V(x)ϕψ−

R^NG(x, η),

for η = (ϕ, ψ) ∈E. Our hypotheses imply that I ∈ C¹(E) and a standard argument shows that its critical points are weak solutions of (ES).

Setting

E⁺=H⁺×H⁻, E⁻=H⁻×H⁺, then for anyz= (u, v)∈E, we have

z=z⁺+z⁻, wherez⁺= (u⁺, v⁻), z⁻= (u⁻, v⁺).

Clearly,E⁺andE⁻are orthogonal with respect to the inner products (·,·)_L²_×L² and (·,·). HenceE=E⁺⊕E⁻. Now we introduce a change of variable ⎧

⎪⎨

⎪⎩

ϕ= u+v

√2 , ψ=u√−v

2 ,

(2.2) and set H(x, z) = H(x, u, v) := G(x,^u+v√₂,^u−v√₂ ), where here and in what follows we write z = (u, v) and

|z|= (|u|²+|v|²)^1/2.

The assumptions onGimply that H satisfies

(H₀) H ∈C¹(R^N ×R²,[0,∞)) is 1-periodic in eachx_i fori= 1, . . . , N;

(H₁) H(x, z) =o(|z|²) as|z| →0.

(H₂) |Hz(x, z)−G_∞(x)z|/|z| →0 as|z| → ∞, whereG_∞ is given in (G₂);

(H₃) G₀:= inf

x∈R^NG_∞(x)>Λ;

(H₄) ˆH(x, z)>0 ifz= 0, and ˆH(x, z)→ ∞as|z| → ∞;

(H₄) there existsδ₀∈(0,Λ₀) such that ˆH(x, z)≥δ₀whenever|H_z(x, z)| ≥(Λ₀−δ₀)|z|; (H₅) ˆH(x, z)≥0, and there existsδ₁>0 such that ˆH(x, z)>0 if 0<|z| ≤δ₁.

It follows from (2.1) and (2.2) that

R^N∇ϕ∇ψ+V(x)ϕψ= 1 2

R^N(|∇u|²+V(x)u²− |∇v|²−V(x)v²)

= 1

2(u⁺²_H− u⁻²_H− v⁺²_H+v⁻²_H)

= 1

2(z⁺²− z⁻²).

Thus, we have an equivalent functional

Φ(z) = Φ(u, v) = 1

2(z⁺²− z⁻²)−Ψ(z), (2.3)

where Ψ(z) =

R^NH(x, z). It is obvious thatz= (u, v) is a critical point of Φ if and only if ((u+v)/√ 2,(u− v)/√

2) is a critical point ofI. In what follows, we shall seek for the critical points of Φ under the assumptions onH. The functional Φ is strongly indefinite; such type of functionals have appeared extensively in the study of differential equationsviacritical point theory, see for example [19,26,27] and the references therein.

3. Linking structure

In this section, we discuss the linking structure of Φ.

Lemma 3.1. Suppose(H₀)–(H₂) are satisfied. Then there is aρ >0 such thatκ:= inf Φ(∂B_ρ∩E⁺)>0.

Proof. Observe that, givenε >0, there isC_ε>0 such that

|H_z(x, z)| ≤ε|z|+C_ε|z|^p−1 (3.1)

and

|H(x, z)| ≤ε|z|²+C_ε|z|^p (3.2) for all (x, z), where p >2. Now, the conclusion follows in a standard way.

By (H₃), we can take a numberγ such that

Λ< γ < G₀. (3.3)

Since σ(S) is absolutely continuous, the subspace Y₁ := (E_γ −E₀)L² and Y₂ := (E₀−E_−γ)L² are infinite dimensional subspaces ofH⁺ andH⁻, respectively. Recall that{E_λ}_λ∈Ris the spectral family ofS. Then

Λ|u|²₂≤ u²_E0≤γ|u|²₂ for allu∈Y₁, Λ|v|²₂≤ v²_E0≤γ|v|²₂ for allv∈Y₂. SetW₀:=Y₁×Y₂, thenW₀ is an infinite dimensional subspace ofE⁺ and

Λ|w|²₂≤ w²_E0≤γ|w|²₂ for allw∈W₀. (3.4) Let{α_n} and{β_n}be two sequences so that

Λ =α₀< α₁< α₂< . . .≤γ,

Λ =¯ β₀> β₁> β₂> . . .≥max{−γ,infσ(S)}.

For each n ∈ N, take an element e_n ∈ (E_αn−E_αn−1)L² with en = 1 and define Y₁ⁿ := span{e1, . . . , e_n}.

Similarly, takef_n∈(E_βn−1−E_βn)L² withf_n= 1 and defineY₂ⁿ:= span{f₁, . . . , f_n}. ThenW_n :=Y₁ⁿ×Y₂ⁿ is a increasing sequence of finite dimensional subspaces ofE⁺. For any subspaceW_n ofW₀setE_n=E⁻⊕W_n. Lemma 3.2. Let (H₀) and (H₂)–(H₃) be satisfied andρ > 0 be given by Lemma 3.1. Then sup Φ(E_n)<∞, and there is a sequenceR_n>0such that sup Φ(E_n\B_n)<inf Φ(B_ρ), whereB_n:={z∈E_n :z ≤R_n}. Proof. It is sufficient to prove that Φ(z)→ −∞inE_n asz → ∞. If not, then there areM >0 and{z_j} ⊂E_n with z_j → ∞such that Φ(z_j)≥ −M for allj. Denote y_j :=z_j/z_j, passing to a subsequence if necessary, y_j y,y_j⁻ y⁻ andy⁺_j →y⁺. Since Ψ(z)≥0,

1

2(y_j⁺²− y_j⁻²)≥1

2(y_j⁺²− y_j⁻²)−Ψ(z_j) z_j²

=Φ(z_j)

z_j² ≥ −M z_j²,

(3.5)

from where it follows that

1

2y⁻_j²≤ 1

2y_j⁺²+ M

z_j²· (3.6)

We claim that y⁺ = 0. Indeed, if not, (3.6) yields that y_j⁻ → 0. Thus y_j → 0, which contradicts with y_j= 1. Since

y⁺²− y⁻²−

R^NG_∞(x)|y|²≤ y⁺²− y⁻²−G₀|y|²₂

≤ −(G0−γ)|y⁺|²₂<0,

then there exists Ω⊂R^N such that

y⁺²− y⁻²−

ΩG_∞(x)|y|²<0. (3.7)

Setting R(x, z) :=H(x, z)−¹₂G_∞(x)|z|², then|R(x, z)| ≤ c|z|² for some c >0, R(x, z)/|z|² →0 as |z| → ∞ uniformly inx. By Lebesgue’s dominated convergence theorem we have

j→∞lim

Ω

|R(x, z_j)|

z_j² = lim

j→∞

Ω

|R(x, z_j)|

|z_j|² |y_j|²= 0. (3.8)

Thus (3.5), (3.7)–(3.8) imply that 0≤ lim

j→∞

1

2(y_j⁺²− y⁻_j²)−

Ω

H(x, z_j) zj²

= lim

j→∞

1

2(y_j⁺²− y⁻_j²)−1 2

ΩG_∞(x)|yj|²−

Ω

R(x, z_j) z_j²

≤1

2(y⁺²− y⁻²−

ΩG_∞(x)|y|²)<0.

Now the desired conclusion follows from this contradiction.

As a consequence, we have:

Lemma 3.3. Let(H₀)and(H₂)–(H₃)be satisfied andκ >0be given by Lemma3.1. Then lettinge∈W₀ with e= 1, there isR₁> ρsuch thatΦ|_∂Q ≤κ, whereQ:={z=z⁻+se:z⁻∈E⁻, s≥0,z ≤R₁}.

4. The (C)

-sequence

Lemma 4.1. Suppose that(H₀)–(H₂)and(H₄)or(H₄)are satisfied. Then any(C)_c-sequence ofΦis bounded.

Proof. Let {z_j} be such that Φ(z_j) → c and (1 +z_j)Φ(z_j) → 0. Suppose to the contrary that {z_j} is unbounded. Settingy_j:=z_j/z_j, theny_j= 1. Without loss of generality, we can assume thaty_j y inE.

Observe that forj large

C≥Φ(z_j)−1

2Φ(z_j)z_j=

R^N

H(x, zˆ _j), (4.1)

and

Φ(z_j)(z⁺_j −z⁻_j) =z_j²−

R^N

H_z(x, z_j)(z_j⁺−z_j⁻)

=z_j²

1−

R^N

H_z(x, z_j)(y⁺_j −y⁻_j ) zj

·

(4.2)

Suppose that (H₄) holds. Set forr≥0, g(r) := inf

H(x, z)|xˆ ∈R^N andz∈R² with|z| ≥r ·

By (H₄),g(r)→ ∞asr→ ∞.

For 0≤a < b, let

Ω_j(a, b) =

x∈R^N|a≤ |z_j(x)|< b and

C_a^b= inf

Hˆ(x, z)

|z|² |x∈R^N witha≤ |z(x)| ≤b

·

One has

Hˆ(x, z_j(x))≥C_a^b|z_j(x)|² for allx∈Ω_j(a, b).

It follows from (4.1) that C≥

Ωj(0,a)

Hˆ(x, z_j) +

Ωj(a,b)

H(x, zˆ _j) +

Ωj(b,∞)

Hˆ(x, z_j)

≥

Ωj(0,a)

Hˆ(x, z_j) +C_a^b

Ωj(a,b)|zj|²+g(b)|Ωj(b,∞)|.

(4.3)

Using (4.3) one has

|Ω_j(b,∞)| ≤ C

g(b) →0 (4.4)

as b→ ∞uniformly inx, and for any fixed 0< a < b,

Ωj(a,b)|y_j|²= 1 z_j²

Ωj(a,b)|z_j|²≤ C

C_a^bz_j² →0 (4.5)

as j→ ∞. It follows from (4.4) that, for anys∈[2,2^∗)

Ωj(b,∞)|y_j|^s≤

Ωj(b,∞)|y_j|^2∗ ₂^s_∗

|Ω_j(b,∞)|^2∗−s2∗ ≤c|Ω_j(b,∞)|^2∗−s2∗ →0 (4.6)

as b→ ∞uniformly inj. In virtue of (4.5), for anys∈[2,2^∗), there holds

Ωj(a,b)|y_j|^s≤c

Ωj(a,b)|y_j|²

₍₂^∗_−s)/(2^∗₋₂₎

→0as j→ ∞. (4.7)

From (4.2),

R^N

H_z(x, z_j)(y⁺_j −y⁻_j )|yj|

|z_j| →1. (4.8)

Let 0< ε <1/3. By (H₁) there isa_ε>0 such that

|Hz(x, z)|< ε c|z|

for all|z| ≤a_ε. Consequently,

Ωj(0,aε)

H_z(x, z_j)(y⁺_j −y⁻_j)|yj|

|z_j| ≤

Ωj(0,aε)

ε

c|y⁺_j −y⁻_j||yj|

≤ε

c|yj|²₂< ε

(4.9)

for allj.

By (H₁) and (H₂), there is someC >0 such that

|Hz(x, z)| ≤C|z| (4.10)

for all (x, z). By (4.6) and H¨older inequality, we can take largeb_εsuch that

Ωj(bε,∞)

H_z(x, z_j)(y⁺_j −y⁻_j )|y_j|

|zj| ≤C

Ωj(bε,∞)|y_j⁺−y_j⁻||y_j|

≤C|Ω_j(b_ε,∞)|^N1

Ωj(bε,∞)|y⁺_j −y⁻_j|^{2 1}₂

Ωj(bε,∞)|y_j|^2∗ ₂¹_∗

≤ε (4.11)

for allj. By (4.5) there isj₀ such that

Ωj(aε,bε)

H_z(x, z_j)(y_j⁺−y_j⁻)|y_j|

|zj| ≤C

Ωj(aε,bε)|y⁺_j −y⁻_j||y_j|

≤C|y_j|2

Ωj(aε,bε)|y_j|²¹₂

≤ε

(4.12)

for allj≥j₀. By (4.9) and (4.10)–(4.12), one has lim sup

n→∞

R³

H_z(x, z_j)(z_j⁺−z_j⁻)

zj² ≤3ε <1, which contradicts (4.8).

In the case that (H₄) holds. By the Lions’ concentration compactness principle, only two cases needed to be considered: {yj} is vanishing or{yj} is nonvanishing.

If {y_j} is vanishing, then by the vanishing lemma we havey_j →0 in L^p(R^N)×L^p(R^N) for p∈(2,2^∗). In virtue of (H₄), set

Ω_j:= x∈R^N||H_z(x, z_j(x))|

|z_j(x)| ≤Λ₀−δ₀

· Then Λ₀|y_j|²₂≤ y_j²= 1 and we have

Ωj

H_z(x, z_j)(y⁺_j −y⁻_j) z_j

=

Ωj

H_z(x, z_j)(y⁺_j −y⁻_j)|yj|

|z_j|

≤(Λ₀−δ₀)|y_j|²₂

≤ Λ₀−δ₀ Λ₀ for allj. This, jointly with (4.8), implies that for Ω^c_j=R^N\Ωj

j→∞lim

Ω^c_j

H_z(x, z_j)(y_j⁺−y_j⁻)|y_j|

|z_j| >1−Λ₀−δ₀ Λ₀ = δ₀

Λ₀· On the other hand, by (4.10) there holds for an arbitrarily fixeds∈(2,2^∗),

Ω^c_j

H_z(x, z_j)(y_j⁺−y_j⁻)|y_j|

|zj| ≤C

Ω^c_j|y⁺_j −y⁻_j||y_j|

≤C|yj|2|Ω^c_j|^(s−2)/2s|yj|s.

Since|y_j|_s→0 and|y_j|₂ is bounded, one has|Ω^c_j| → ∞. From (H₄), ˆH(x, z)≥δ₀ on Ω^c_j and hence

R^N

Hˆ(x, z_j)≥

Ω^c_j

Hˆ(x, z_j)≥δ₀|Ω^c_j| → ∞,

contrary to (4.1).

If{y_j} is nonvanishing,i.e., there existα >0,R <∞and{a_j} ⊂R^N such that lim inf

j→∞

B(aj,R)|y_j|²≥α.

Set ˜z_j(x) =z_j(x+a_j), ˜y_j(x) =y_j(x+a_j) and η_j(x) =η(x+a_j) for eachη = (ϕ, ψ)∈C₀^∞(R^N)×C₀^∞(R^N).

Now from (H₂) there holds

Φ(z_j)η_j = (z⁺_j −z⁻_j, η_j)−(G_∞(x)z_j, η_j)_L2×L²−

R^N

R_z(x, z_j)η_j

=z_j

(y⁺_j −y⁻_j, η_j)−(G_∞(x)y_j, η_j)_L2×L²−

R^NR_z(x, z_j)η_j|yj|

|z_j|

=z_j

(˜y⁺_j −y˜⁻_j, η)−(G_∞(x)˜y_j, η)_L2×L²−

R^NR_z(x,z˜_j)η|y˜_j|

|˜z_j|

and hence

(˜y⁺_j −y˜⁻_j, η)−(G_∞(x)˜y_j, η)_L2×L²−

R^N

R_z(x,z˜_j)η|y˜_j|

|˜z_j| =o(1). (4.13) Sincey˜_j=y_j= 1, up to a subsequence we may assume that ˜y_jy˜in E, ˜y_j →y˜inL²_loc(R^N)×L²_loc(R^N) and ˜y_j(x)→y(x) a.e. in˜ R^N. Clearly, ˜y= 0. Observe that

R^N

R_z(x,z˜_j)η|y˜_j|

|˜z_j| ≤

R^N

R_z(x,z˜_j)η|y˜|

|˜z_j|

+

R^N

R_z(x,z˜_j)η|y˜_j−y˜|

|˜z_j|

=I+II.

Since ^Rz(x,˜_|˜z^zj)η|˜y|

j| →0 a.e. in R^N, it follows from the dominated convergence theorem that I → 0. By local compactness of embedding, one has

II =

suppη

R_z(x,z˜_j)η|˜y_j−y|˜

|˜z_j|

≤C|η|2|˜y_j−y|˜_L²₍suppη)→0.

Combining the above two estimates forI andII, we have

R^N

R_z(x,z˜_j)η|y˜_j|

|˜z_j| →0, and letting j→ ∞in (4.13) we get

(˜y⁺−y˜⁻, η)−(G_∞(x)˜y, η)_L2×L²= 0, for eachη= (ϕ, ψ)∈C₀^∞(R^N). (4.14) Let ˜y= (ζ, ξ), then ˜y⁺−y˜⁻= (ζ⁺−ζ⁻, ξ⁻−ξ⁺). Thus from (4.14) we have

(ζ⁺−ζ⁻, ϕ)_H+ (ξ⁻−ξ⁺, ψ)_H−(G_∞(x)ζ, ϕ)_L2−(G_∞(x)ξ, ψ)_L2 = 0,

which implies that

−Δζ+V(x)ζ=G_∞ζ, Δξ+V(x)ξ=G_∞ξ.

Hence ζ is an eigenfunction of S₁ := −Δ + (V −G_∞) and ξ is an eigenfunction of S₂ := −Δ + (G_∞−V), which contradicts with the fact that S_i has only continuous spectrum for i= 1,2 since V −G_∞ is 1-periodic.

Therefore{z_j}is bounded in E.

Let{zj} ⊂E be a (C)_c-sequence of Φ, by Lemma3.1, it is bounded, up to a subsequence, we may assume z_j zin E, z_j→z inL^p_locforp∈[2,2^∗) andz_j(x)→z(x) a.e. onR^N. Plainly,z is a critical point of Φ.

Recall that a mappingf from a Banach spaceX to another Banach spaceY is called a BL-split, if for every sequence{x_n} inX withx_n xit holds thatf(x_n)−f(x_n−x)→f(x) inY (see Ackermann [2]). We adopt here a cut-off technique developed in Ackermann [1,2]. Let η : [0,∞)→[0,1] be a smooth function satisfying η(s) = 1 if s ≤ 1, η(s) = 0 if s ≥2. Define ˜z_j(x) = η(2|x|/j)z(x), then ˜z_j → z in E. In a similar way to Lemma 3.2 in Ackermann [1] (see also Lem. 4.4 in Ding and Jeanjean [15]), we can obtain the following:

Lemma 4.2. Under the assumptions of Theorem 1.1,Ψ(·)andΨ(·)are both BL-splits.

LetK:={z∈E|Φ(z) = 0, z= 0} be the set of nontrivial critical points of Φ.

Lemma 4.3. Under the assumptions of Lemma 4.1, the following two conclusions hold (1)ν := inf{z:z∈ K}>0;

(2)θ:= inf{Φ(z)|z∈ K}>0.

Proof. (1) For anyz∈ K, there holds

0 = Φ(z)(z⁺−z⁻) =z²−

R^NH_z(x, z)(z⁺−z⁻), jointly with (3.1), which implies that

z²≤ε|z|²₂+C_ε|z|^p_p≤cεz²₂+cC_εz^p, wherep∈(2,2^∗). Chooseεsmall enough, hence

0<

1−cε cC_ε

_p−2¹

≤ z for eachz∈ K.

(2) Suppose to the contrary that there exist a sequence {z_j} ∈ K such that Φ(z_j)→ 0. By (1),z_j ≥ν. Clearly, {z_j} is a (C)₀-sequence of Φ, and hence is bounded by Lemma 4.1. Moreover,{z_j} is nonvanishing.

By the invariance under translation of Φ, we can assume, up to a translation, thatz_j z∈ K. Sincez= (u, v) is a solution of (ES),|z(x)| →0 as|x| → ∞. Thus there is a bounded domain Ω⊂R^N with positive measure such that 0<|z(x)|< δ₁ forx∈Ω by (H₅). Then

o(1) = Φ(z_j) = Φ(z_j)−1

2Φ(z_j)z_j =

R^N

Hˆ(x, z_j)≥

Ω

H(x, zˆ _j), letting j→ ∞yields

0≥ lim

j→∞

Ω

H(x, zˆ _j)≥

Ω

Hˆ(x, z)>0.

This ends the proof.

In the following lemma we discuss further the (C)_c-sequence. Let [l] denote the integer part of l ∈R. The following lemma is standard (see Coti-Zelati and Rabinowitz [9,10] and S´er´e [24]).

Lemma 4.4. Under the assumptions of Theorem 1.1, let{z_j} ⊂E be a(C)_c-sequence of Φ. Then either (i)z_j →0 (and hencec= 0), or

(ii)c≥θ and there exist a positive integerl≤[c/θ],y₁, . . . , y_l∈ K and sequences aⁱ_j

⊂Z^N,i= 1,2, . . . , l, such that, after extraction of a subsequence of {z_j},

z_j−^l

i=1

aⁱ_j∗y_i→0,

l i=1

Φ(y_i) =c and for i=k,

|aⁱ_j−a^k_j| → ∞ asj→ ∞.

5. Proof of Theorem 1.1

In this section we prove our Theorem1.1. First, we recall some terminology from Bartsch and Ding [7]. Let Ebe a Banach space with direct sumE=X⊕Y and corresponding projectionsP_X, P_Y ontoX, Y. LetS ⊂X^∗ be a dense subset, for eachs∈ S there is a semi-norm onE defined by

p_s:E→R, p_s(u) =|s(x)|+yforu=x+y∈E.

We denote by TS the topology induced by the semi-norm family{ps}, and byw^∗ denote the weak^∗-topology on E^∗. Now, some notations are needed. For a functional Φ ∈ C¹(E,R) we write Φa = {u∈E|Φ(u)≥a}, Φ^b = {u∈E|Φ(u)≤b} and Φ^b_a = Φ_a∩Φ^b. Recall that a set A ⊂ E is said to be a (C)_c-attractor if for any ε, δ > 0 and any (C)_c-sequence {z_j} there is j₀ such that z_j ∈ U_ε(A ∩Φ^c+δ_c−δ) for j ≥ j₀. Given an interval I ⊂R, A is said to be a (C)_I-attractor if it is a (C)_c-attractor for all c ∈ I. Φ is said to be weakly sequentially lower semicontinuous if for any u_j u in E one has Φ(u) ≤lim inf

j→∞ Φ(u_j), and Φ is said to be weakly sequentially continuous if lim

j→∞Φ(u_j)w= Φ(u)wfor eachw∈E. Suppose (Φ₀) for anyc∈R, Φ_c isT_S-closed, and Φ: (Φ_c,T_S)→(E^∗, w^∗) is continuous;

(Φ₁) for anyc >0, there existsξ >0 such thatu< ξP_Yu for allu∈Φ_c;

(Φ₂) there existsρ >0 such thatκ:= inf Φ(S_ρ∩Y)>0, whereS_ρ:={u∈E :u=ρ};

(Φ₃) there exists a finite-dimensional subspaceY₀⊂Y andR > ρsuch that we have forE₀ :=X⊕Y₀ and B₀:={u∈E₀:u ≤R}that ¯c:= sup Φ(E₀)<∞and sup Φ(E₀\B₀)<inf Φ(B_ρ∩Y);

(Φ₄) there is an increasing sequenceY_n⊂Y of finite-dimensional subspaces and a sequence{Rn}of positive numbers such that, lettingE_n =X⊕Y_n and B_n =B_Rn∩E_n, sup Φ(E_n)<∞and sup Φ(E_n\B_n)<

inf Φ(B_ρ∩Y);

(Φ₅) for any intervalI⊂(0,∞) there is a (C)_I-attractorAwithP_XAbounded and inf{P_Y(z−w):z, w∈A, P_Y(z−w)= 0}>0.

Now we state two critical point theorems which will be used later (see Bartsch-Ding [7]).

Theorem 5.1. Let (Φ₀)–(Φ₂)be satisfied and suppose there are R > ρ >0 ande∈Y with e= 1such that sup Φ(∂Q) ≤κ where Q:= {u=x+te:x∈X, t≥0,u< R}. Then Φ has a (C)_c-sequence with κ ≤c ≤ sup Φ(Q).