Skip to content

Advertisement

Open Access

Semi-classical solutions of perturbed elliptic system with general superlinear nonlinearity

Boundary Value Problems20142014:208

https://doi.org/10.1186/s13661-014-0208-1

Received: 1 June 2014

Accepted: 21 August 2014

Published: 25 September 2014

Abstract

This paper is concerned with the following perturbed elliptic system: ε 2 u + V ( x ) u = W v ( x , u , v ) , x R N , ε 2 v + V ( x ) v = W u ( x , u , v ) , x R N , u , v H 1 ( R N ) , where V C ( R N , R ) and W C 1 ( R N × R 2 , R ) . Under some mild conditions on the potential V and nonlinearity W, we establish the existence of nontrivial semi-classical solutions via variational methods, provided that 0 < ε ε 0 , where the bound ε 0 is formulated in terms of N, V, and W.

MSC: 35J10, 35J20.

Keywords

semi-classical solutionsperturbed elliptic systemgeneralized linking theorems

1 Introduction

The goal of this paper is to establish the existence of semi-classical solutions to the following perturbed elliptic system of Hamiltonian form:
{ ε 2 u + V ( x ) u = W v ( x , u , v ) , x R N , ε 2 v + V ( x ) v = W u ( x , u , v ) , x R N , u , v H 1 ( R N ) ,
(1.1)

where u , v : R N R , ε > 0 is a small parameter, and V : R N R and W : R N × R 2 R satisfy the following basic assumptions, respectively:

(V0) V C ( R N ) and there exists a b > 0 such that the set V b : = { x R N : V ( x ) < b } has finite measure;

(V1) V ( x ) min V = 0 ;

(W1) W C 1 ( R N × R 2 ) , and there exist constants p 0 ( 2 , 2 ) and C 0 > 0 such that
| W z ( x , z ) | C 0 ( 1 + | z | p 0 1 ) , ( x , z ) R N × R 2 , z = ( u , v ) ;

here and in the sequel 2 : = 2 N / ( N 2 ) if N 3 and 2 : = + if N = 1 or 2;

(W2) | W z ( x , z ) | = o ( | z | ) , as | z | 0 , uniformly in x R N .

For the case ε = 1 , the interest in the study of various qualitative properties of the solutions has steadily increased in recent years. In a bounded smooth domain Ω R N , similar systems have been extensively studied; see, for instance, [1]–[6] and the references therein. The problem set on the whole space R N was considered recently in some works. The first difficulty of such a type of problem is the lack of the compactness of the Sobolev embedding. A usual way to recover this difficulty is choosing suitable working space which has compact embedding property, for example, the radially symmetric functions space; see [7]–[9]. The second difficulty is that the energy functional is strongly indefinite different from the single equation case, and so the dual variational methods are involved to avoid this difficulty; see [10], [11]. Recently, with the aid of the linking arguments in [12]–[14], the existence of solutions or multiple solutions were obtained with periodic potential and nonlinearity, see [15]–[19] and the references therein.

For the problem with a small parameter ε > 0 , it is called the semi-classical problem, which describes the transition between of quantum mechanics and classical mechanics with the parameter ε goes to zero. There is much literature dealing with the existence of semi-classical solutions to the single particle equation
ε 2 u + V ( x ) u = g ( x , u ) , x R N , u H 1 ( R N ) ,
(1.2)

under various hypotheses on the potential V and the nonlinearity g; for example, see [20]–[27] and references therein. In a very recent paper [28], Lin and Tang developed a direct and simple approach to show the existence of semi-classical solutions for the single particle equation (1.2) with V satisfying (V0) and (V1). It is well known that the extension of these results of single equation to a system of equations presents some difficulties. One of the main difficulties is that the energy functional associated with (1.1) is strongly indefinite, so the approach used in the single equation is not applicable to system (1.1).

Inspired by the single equation, there are a few works considering the perturbed elliptic systems; see [11], [29]–[35] and references therein. To the best of our knowledge, the first approach to the singular perturbed system in a bounded domain, with Neumann boundary, and V ( x ) 1 appeared in [11] by means of a dual variational formulation of the problem. Moreover, in [33], Sirakov and Soares considered the superquadratic case by using dual variational methods. In [31], Ramos and Tavares considered the following problem:
{ ε 2 u + V ( x ) u = g ( v ) , x Ω , ε 2 v + V ( x ) v = f ( u ) , x Ω , u ( x ) = 0 and v ( x ) = 0 on  Ω ,

where Ω is a domain of R N , f and g are power functions, superlinear but subcritical at infinity. The authors established the existence of positive solutions which concentrate, as ε 0 , on a prescribed finite number of local minimum points of the potential V; also see [30].

Since Kryszewski and Szulkin [6] proposed the generalized linking theorem for the strongly indefinite functionals in 1998, Li and Szulkin [13], Bartsch and Ding [12] gave several weaker versions, which provide another effective way to deal with such problems. With the aid of the generalized linking theorem, Xiao et al.[34] studied the asymptotically quadratic case and obtained the existence of multiple solutions. For the superquadratic case with magnetic potential, we refer the reader to [35] and references therein.

In the aforementioned references, it always was assumed that W satisfies a condition of the type of Ambrosetti-Rabinowitz, that is,

(AR) there is a μ > 2 such that
0 < μ W ( x , z ) W z ( x , z ) z , ( x , z ) R N × R 2 , z 0 ;
(1.3)
together with a technical assumption that there is ν > 2 N / ( N + 2 ) such that
| W z ( x , z ) | ν c [ 1 + W z ( x , z ) z ] , ( x , z ) R N × R 2 ;
(1.4)
or the superquadratic condition
lim | z | | W ( x , z ) | | z | 2 = , uniformly in  x R N
(1.5)

and a condition of the type of Ding-Lee [36];

(DL) W ˜ ( x , z ) : = 1 2 W z ( x , z ) z W ( x , z ) > 0 for z 0 and there exist c 0 > 0 and κ > max { 1 , N / 2 } such that
| W z ( x , z ) | κ c 0 | z | κ W ˜ ( x , z ) for large  | z | .
(1.6)

As is well known, conditions (1.3) and (1.4) have been successfully applied to Hamiltonian systems, to periodic Schrödinger systems, and to diffusion systems; see [37], [38] and so on. We refer the reader to [39]–[43] and the references therein where the condition (AR) was weakened by more general superlinear conditions. Condition (DL) was firstly given for a single Schrödinger equation by Ding and Lee [36]. Soon after, this condition was generalized by Zhang et al.[44].

Observe that conditions (1.5) and W ( x , z ) > 0 , z 0 in (AR) or W ˜ ( x , z ) > 0 , z 0 , in (DL) play an important role in showing that any Palais-Smale sequence or Cerami sequence is bounded in the aforementioned works. However, there are many functions which do not satisfy these conditions, for example,
W ( x , u , v ) = | u + v | ϱ , ϱ ( 2 , 2 )
or
W ( x , u , v ) = ( u + v ) 2 u 2 + v 2 .

Motivated by these works, in the present paper, we shall establish the existence of semi-classical solutions of system (1.1) with a weaker superlinear condition via the generalized linking theorem. To state our results, in addition to the basic hypotheses, we make the following assumptions:

(W3) there exist a 0 > 0 and p ( 2 , 2 ) such that
W ( x , u , v ) a 0 | u + v | p , ( x , u , v ) R N × R 2 ;
(W4) W ˜ ( x , z ) 0 , ( x , z ) R N × R 2 , and there exist R 0 > 0 , a 1 > 0 , and κ > max { 1 , N / 2 } such that
| W u ( x , z ) + W v ( x , z ) | b 3 | z | , ( x , z ) R N × R 2 , | z | R 0
and
| W u ( x , z ) + W v ( x , z ) | κ a 1 | z | κ W ˜ ( x , z ) , ( x , z ) R N × R 2 , | z | R 0 .

In the present paper, we make use of the techniques developed in [28], [45] to obtain the existence of semi-classical solutions for system (1.1) when ε ε 0 , where the bound ε 0 is formulated in terms of N, V, and W.

Since ( p 2 ) N 2 p < 0 , we can choose a h 0 1 such that
( p 2 ) ω N 4 N a 0 2 / ( p 2 ) [ N 3 + 2 ( N + 2 ) 2 ( N + 2 ) p ( 1 2 N ) 2 ] p / ( p 2 ) h 0 [ ( p 2 ) N 2 p ] / ( p 2 ) b ( 2 κ N ) / 2 3 κ 2 a 1 ( γ 2 γ 0 ) N .
(1.7)
Let x 0 R N be such that V ( x 0 ) = 0 . Then we can choose λ 0 > 1 such that
sup λ 1 / 2 | x | 2 h 0 | V ( x 0 + x ) | h 0 2 , λ λ 0 .
(1.8)
Let E be a Hilbert space as defined in Section 2 and λ = ε 2 . Under assumptions (V0), (V1), (W1), and (W2), the functional
Φ λ ( z ) = R N ( u v + λ V ( x ) u v ) d x λ R N W ( x , u , v ) d x , z = ( u , v ) E ,
(1.9)
is well defined. Moreover, Φ λ C 1 ( E , R ) and for all z = ( u , v ) , ζ = ( φ , ψ ) E
Φ λ ( z ) , ζ = R N [ u ψ + v φ + λ V ( x ) ( u ψ + v φ ) ] d x λ R N [ W u ( x , u , v ) φ + W v ( x , u , v ) ψ ] d x .
(1.10)

We are now in a position to state the main results of this paper.

Theorem 1.1

Assume that V and W satisfy (V0), (V1), (W1), (W2), (W3), and (W4). Then for 0 < ε λ 0 1 / 2 , (1.1) has a solution ( u ε , v ε ) such that 0 < Φ ε 1 / 2 ( u ε , v ε ) b ( 2 κ N ) / 2 3 κ 2 a 1 ( γ 2 γ 0 ) N ε N 2 , and
R N W ˜ ( x , u ε , v ε ) d x b ( 2 κ N ) / 2 3 κ 2 a 1 ( γ 2 γ 0 ) N ε N .

Theorem 1.2

Assume that V and W satisfy (V0), (V1), (W1), (W2), (W3), and (W4). Then for λ λ 0 , (2.1) has a solution ( u λ , v λ ) such that 0 < Φ λ ( u λ , v λ ) b ( 2 κ N ) / 2 3 κ 2 a 1 ( γ 2 γ 0 ) N λ 1 N / 2 , and
R N W ˜ ( x , u λ , v λ ) d x b ( 2 κ N ) / 2 3 κ 2 a 1 ( γ 2 γ 0 ) N λ N / 2 .

Before proceeding to the proofs of these theorems, we give two examples to illustrate the assumptions.

Example 1.3

W ( x , u , v ) = h ( x ) | u + v | ϱ satisfies (W1)-(W4), where ϱ ( 2 , 2 ) , h C ( R N ) with inf R N h > 0 .

Example 1.4

W ( x , u , v ) = h ( x ) ( u + v ) 2 2 u 2 + v 2 satisfies (W1)-(W4), where h C ( R N ) with inf R N h > 0 .

The rest of the paper is organized as follows. In Section 2, we provide a variational setting. In Section 3, we give the proofs of our theorems.

2 Variational setting

Letting ε 2 = λ , (1.1) is rewritten as
{ u + λ V ( x ) u = λ W v ( x , u , v ) , x R N , v + λ V ( x ) v = λ W u ( x , u , v ) , x R N , u , v H 1 ( R N ) .
(2.1)
Let
E V = { u H 1 ( R N ) : R N V ( x ) u 2 d x < + } , ( u , v ) λ V = R N [ u v + λ V ( x ) u v ] d x , u , v E V
and
u λ V = { R N [ | u | 2 + λ V ( x ) u 2 ] d x } 1 / 2 , u E V .
Analogous to the proof of [46], Lemma 1], by using (V0), (V1), and the Sobolev inequality, one can demonstrate that there exists a constant γ 0 > 0 independent of λ such that
u H 1 ( R N ) γ 0 u λ V , u E V , λ 1 .
(2.2)
This shows that ( E V , ( , ) λ V ) is a Hilbert space for λ 1 . Furthermore, by virtue of the Sobolev embedding theorem, we have
u s γ s u H 1 ( R N ) γ s γ 0 u λ V , u E V , λ 1 , 2 s 2 ;
(2.3)

here and in the sequel, by s we denote the usual norm in space L s ( R N ) .

Set E = E V × E V , then E is a Hilbert space with the inner product
( z 1 , z 2 ) λ = ( u 1 , u 2 ) λ V + ( v 1 , v 2 ) λ V , z i = ( u i , v i ) E , i = 1 , 2 ,
the corresponding norm is denoted by λ . Then we have
z λ 2 = u λ V 2 + v λ V 2 , z = ( u , v ) E
(2.4)
and
z s s = R N ( u 2 + v 2 ) s / 2 d x 2 ( s 2 ) / 2 ( u s s + v s s ) 2 ( s 2 ) / 2 ( γ s γ 0 ) s ( u λ V s + v λ V s ) 2 ( s 2 ) / 2 ( γ s γ 0 ) s ( u λ V 2 + v λ V 2 ) s / 2 = 2 ( s 2 ) / 2 ( γ s γ 0 ) s z λ s , s ( 2 , 2 ] , z = ( u , v ) E .
(2.5)
Let
E = { ( u , u ) : u E V } , E + = { ( u , u ) : u E V } .
For any z = ( u , v ) E , set
z = ( u v 2 , v u 2 ) , z + = ( u + v 2 , u + v 2 ) .
It is obvious that z = z + z + , z and z + are orthogonal with respect to the inner products ( , ) L 2 and ( , ) λ . Thus we have E = E E + . By a simple calculation, one gets
1 2 ( z + λ 2 z λ 2 ) = R N [ u v + λ V ( x ) u v ] d x .
Therefore, the functional Φ λ defined in (1.9) can be rewritten in a standard way
Φ λ ( z ) = 1 2 ( z + λ 2 z λ 2 ) λ R N W ( x , u , v ) d x , z = ( u , v ) E .
(2.6)
Moreover,
Φ λ ( z ) , z = z + λ 2 z λ 2 λ R N [ W u ( x , u , v ) u + W v ( x , u , v ) v ] d x , z = ( u , v ) E .
(2.7)

The following generalized linking theorem provides a convenient approach to get a ( C ) c sequence.

Let X be a Hilbert space with X = X X + and X X + . For a functional φ C 1 ( X , R ) , φ is said to be weakly sequentially lower semi-continuous if for any u n u in X one has φ ( u ) lim inf n φ ( u n ) , and φ is said to be weakly sequentially continuous if lim n φ ( u n ) , v = φ ( u ) , v for each v X .

Lemma 2.1

([37], Theorem 4.5], [13], Theorem 2.1])

Let X be a Hilbert space with X = X X + and X X + , and let φ C 1 ( X , R ) be of the form
φ ( u ) = 1 2 ( u + 2 u 2 ) ψ ( u ) , u = u + + u X + X .

Suppose that the following assumptions are satisfied:

(I1) ψ C 1 ( X , R ) is bounded from below and weakly sequentially lower semi-continuous;

(I2) ψ is weakly sequentially continuous;

(I3)there exist r > ρ > 0 and e X + with e = 1 such that
κ : = inf φ ( S ρ ) > sup φ ( Q ) ,
where
S ρ = { u X + : u = ρ } , Q = { s e + v : v X , s 0 , s e + v r } .
Then for some c κ , there exists a sequence { u n } X satisfying
φ ( u n ) c , φ ( u n ) ( 1 + u n ) 0 .

Such a sequence is called a Cerami sequence on the level c, or a ( C ) c sequence.

3 Proofs of the theorems

In this section, we give the proofs of Theorems 1.1 and 1.2.

From now on we assume without loss of generality that x 0 = 0 (see [47]), that is, V ( 0 ) = 0 , then λ 0 > 1 defined by (1.8) satisfies
sup λ 1 / 2 | x | 2 h 0 | V ( x ) | h 0 2 , λ λ 0 .
(3.1)
Let
ϑ ( x ) : = { 1 h 0 , | x | h 0 , h 0 N 1 1 2 N [ | x | N ( 2 h 0 ) N ] , h 0 < | x | 2 h 0 , 0 , | x | > 2 h 0 .
(3.2)
Then ϑ H 1 ( R N ) , moreover,
ϑ 2 2 = R N | ϑ ( x ) | 2 d x N 2 ω N ( N + 2 ) ( 1 2 N ) 2 h 0 N 4 ,
(3.3)
ϑ 2 2 = R N | ϑ ( x ) | 2 d x 2 ω N ( 1 2 N ) 2 N h 0 N 2
(3.4)
and
ϑ p p = R N | ϑ ( x ) | p d x ω N N h 0 N p .
(3.5)

Let e λ ( x ) = ( ϑ ( λ 1 / 2 x ) , 0 ) . Then we can prove the following lemma which is very important and crucial.

Lemma 3.1

Suppose that (V0), (V1), (W1), (W2), and (W3) are satisfied. Then
sup { Φ λ ( ζ + s e λ ) : ζ = ( w , w ) E , s 0 } b ( 2 κ N ) / 2 3 κ 2 a 1 ( γ 2 γ 0 ) N λ 1 N / 2 , λ λ 0 .
(3.6)

Proof

Note that e λ + = ( ϑ ( λ 1 / 2 x ) / 2 , ϑ ( λ 1 / 2 x ) / 2 ) , we have
e λ + λ 2 = 1 2 R N ( | ϑ ( λ 1 / 2 x ) | 2 + λ V ( x ) | ϑ ( λ 1 / 2 x ) | 2 ) d x = 1 2 λ 1 N / 2 R N ( | ϑ | 2 + V ( λ 1 / 2 x ) | ϑ | 2 ) d x 1 2 λ 1 N / 2 ( ϑ 2 2 + ϑ 2 2 sup λ 1 / 2 | x | 2 h 0 | V ( x ) | ) 1 2 λ 1 N / 2 ( ϑ 2 2 + h 0 2 ϑ 2 2 ) , λ λ 0 .
(3.7)
It follows from (W3), (1.7), (1.9), (3.3), (3.4), (3.5), and (3.7) that
Φ λ ( ζ + s e λ ) = 1 2 ( s 2 e λ + λ 2 ζ + s e λ λ 2 ) λ R N W ( x , w + s ϑ ( λ 1 / 2 x ) , w ) d x s 2 2 e λ + λ 2 a 0 λ s p R N | ϑ ( λ 1 / 2 x ) | p d x λ 1 N / 2 { s 2 4 ( ϑ 2 2 + h 0 2 ϑ 2 2 ) a 0 s p ϑ p p } λ 1 N / 2 [ s 2 4 ( N 3 + 2 ( N + 2 ) N ( N + 2 ) ( 1 2 N ) 2 ) ω N h 0 N 4 a 0 ω N N s p h 0 N p ] ( p 2 ) ω N 4 N a 0 2 / ( p 2 ) [ N 3 + 2 ( N + 2 ) 2 ( N + 2 ) p ( 1 2 N ) 2 ] p / ( p 2 ) h 0 [ ( p 2 ) N 2 p ] / ( p 2 ) λ 1 N / 2 b ( 2 κ N ) / 2 3 κ 2 a 1 ( γ 2 γ 0 ) N λ 1 N / 2 , s 0 , λ λ 0 , ζ = ( w , w ) E .
(3.8)

Now the conclusion of Lemma 3.1 follows by (3.8). □

Applying Lemma 2.1, by standard arguments (see, e.g., [45]), we can prove the following lemma.

Lemma 3.2

Suppose that (V0), (V1), (W1), (W2), and (W3) are satisfied. Then there exist a constant c λ ( 0 , sup { Φ λ ( ζ + s e λ ) : ζ = ( w , w ) E , s 0 } ] and a sequence { z n } = { ( u n , v n ) } E satisfying
Φ λ ( z n ) c λ , Φ λ ( z n ) ( 1 + z n ) 0 .
(3.9)

Lemma 3.3

Suppose that (V0), (V1), (W1), (W2), (W3), and (W4) are satisfied. Then any sequence { z n } = { ( u n , v n ) } E satisfying (3.9) is bounded in E.

Proof

By virtue of (W3), (2.6), and (3.9), one gets
2 c λ + o ( 1 ) = z n + λ 2 z n λ 2 2 λ R N W ( x , u n , v n ) d x z n + λ 2 z n λ 2 2 a 0 λ R N | u n + v n | p d x .
(3.10)
To prove the boundedness of { z n } , arguing by contradiction, suppose that z n λ . Let ξ n = z n / z n λ = ( φ n , ψ n ) , then ξ n λ = 1 . If
δ : = lim sup n sup y R N B ( y , 1 ) | ξ n + | 2 d x = 0 ,
then by Lions’ concentration compactness principle [48] or [49], Lemma 1.21], φ n + ψ n 0 in L s ( R N ) for 2 < s < 2 . Hence, it follows from (W4), (2.4), (3.10), and the Hölder inequality that
λ 2 | z n | R 0 | W u ( x , u n , v n ) + W v ( x , u n , v n ) | | u n + v n | d x λ b 6 | z n | R 0 | z n | | u n + v n | d x λ b 6 R N V b | z n | | u n + v n | d x + λ b 6 V b | z n | | u n + v n | d x λ b 6 ( R N V b | z n | 2 d x ) 1 / 2 ( R N V b | u n + v n | 2 d x ) 1 / 2 + λ b [ meas ( V b ) ] 1 / ( N + 1 ) 6 ( V b | z n | 2 ( N + 1 ) / N d x ) N / 2 ( N + 1 ) × ( V b | u n + v n | 2 ( N + 1 ) / N d x ) N / 2 ( N + 1 ) 1 6 z n λ u n + v n λ V + λ b [ meas ( V b ) ] 1 / ( N + 1 ) 6 z n 2 ( N + 1 ) / N u n + v n 2 ( N + 1 ) / N = 1 6 z n λ u n + v n λ V + λ b [ meas ( V b ) ] 1 / ( N + 1 ) 6 ξ n 2 ( N + 1 ) / N × φ n + ψ n 2 ( N + 1 ) / N z n λ 2 [ 1 3 + o ( 1 ) ] z n λ 2 .
(3.11)
From (2.6), (2.7), and (3.9), one has
c λ + o ( 1 ) = λ R N W ˜ ( x , u n , v n ) d x .
(3.12)
Let κ = κ / ( κ 1 ) , then 2 < 2 κ < 2 . Hence, by virtue of (W4), (3.12), and the Hölder inequality, one gets
λ 2 | z n | R 0 | W u ( x , u n , v n ) + W v ( x , u n , v n ) | | u n + v n | z n λ 2 d x = λ 2 | z n | R 0 | W u ( x , u n , v n ) + W v ( x , u n , v n ) | | ξ n | | φ n + ψ n | | z n | d x λ 2 ( | z n | R 0 | W u ( x , u n , v n ) + W v ( x , u n , v n ) z n | κ d x ) 1 / κ ( | z n | R 0 | ξ n | 2 κ d x ) 1 / 2 κ × ( | z n | R 0 | φ n + ψ n | 2 κ d x ) 1 / 2 κ λ 2 ( a 1 | z n | R 0 W ˜ ( x , u n , v n ) d x ) 1 / κ ξ n 2 κ φ n + ψ n 2 κ λ ( κ 1 ) / κ ( c λ a 1 ) 1 / κ ξ n 2 κ φ n + ψ n 2 κ = o ( 1 ) .
(3.13)
Combining (3.11) with (3.13) and using (1.10), (3.9), and (3.10), we have
1 2 + o ( 1 ) z n + λ 2 Φ λ ( z n ) , z n + z n λ 2 = λ 2 R N [ W u ( x , u n , v n ) + W v ( x , u n , v n ) ] ( u n + v n ) z n λ 2 d x = λ 2 | z n | R 0 [ W u ( x , u n , v n ) + W v ( x , u n , v n ) ] ( u n + v n ) z n λ 2 d x + λ 2 | z n | > R 0 [ W u ( x , u n , v n ) + W v ( x , u n , v n ) ] ( u n + v n ) z n λ 2 d x 1 3 + o ( 1 ) .
(3.14)

This contradiction shows that δ > 0 .

Going if necessary to a subsequence, we may assume the existence of k n Z N such that B 1 + N ( k n ) | ξ n + | 2 d x > δ 2 . Let ζ n ( x ) = ξ n ( x + k n ) . Then
B 1 + N ( 0 ) | ζ n + | 2 d x > δ 2 .
(3.15)
Now we define z ˜ n ( x ) = ( u ˜ n , v ˜ n ) = z n ( x + k n ) , then z ˜ n / z n λ = ζ n and ζ n H 1 ( R N ) 2 = ξ n H 1 ( R N ) 2 . Passing to a subsequence, we have ζ n ζ in E, ζ n ζ in L loc s ( R N ) , 2 s < 2 , and ζ n ζ a.e. on R N . Obviously, (3.15) implies that ζ + 0 . For a.e. x { y R N : ζ + ( y ) 0 } : = Ω , we have lim n | u ˜ n ( x ) + v ˜ n ( x ) | = . Hence, it follows from (2.6), (3.9), (W3), and Fatou’s lemma that
0 = lim n c + o ( 1 ) z n λ 2 = lim n Φ λ ( z n ) z n λ 2 = lim n [ 1 2 ( ξ n + λ 2 ξ n λ 2 ) λ R N W ( x , u n , v n ) z n λ 2 d x ] = lim n [ 1 2 ( ξ n + λ 2 ξ n λ 2 ) λ R N W ( x + k n , u ˜ n , v ˜ n ) | z ˜ n + | 2 | ζ n + | 2 d x ] = lim n [ 1 2 ( ξ n + λ 2 ξ n λ 2 ) 2 λ R N W ( x + k n , u ˜ n , v ˜ n ) | u ˜ n + v ˜ n | 2 | ζ n + | 2 d x ] 1 2 2 λ Ω lim inf n W ( x + k n , u ˜ n , v ˜ n ) | u ˜ n + v ˜ n | 2 | ζ n + | 2 d x = .

This contradiction shows that z n λ is bounded. □

Proof of Theorem 1.2

Applying Lemmas 3.1, 3.2, and 3.3, we deduce that there exists a bounded sequence { z n } = { ( u n , v n ) } E satisfying (3.9) and (3.10) with
c λ b ( 2 κ N ) / 2 3 κ 2 a 1 ( γ 2 γ 0 ) N λ 1 N / 2 , λ λ 0 .
(3.16)

Going if necessary to a subsequence, we can assume that z n z λ = ( u λ , v λ ) in ( E , λ ) , and Φ λ ( z n ) 0 . Next, we prove that z λ 0 .

Arguing by contradiction, suppose that z λ = 0 , i.e. z n 0 in E, and so u n 0 , v n 0 in L loc s ( R N ) , 2 s < 2 , and u n 0 , v n 0 a.e. on R N . Since V b is a set of finite measure, we have
u n + v n 2 2 = R N V b | u n + v n | 2 d x + V b | u n + v n | 2 d x 1 λ b u n + v n λ V 2 + o ( 1 )
(3.17)
and
z n 2 2 = R N V b ( u n 2 + v n 2 ) d x + V b ( u n 2 + v n 2 ) d x 1 λ b z n λ 2 + o ( 1 ) .
(3.18)
For s ( 2 , 2 ) , it follows from (2.3), (2.5), (3.17), (3.18), and the Hölder inequality that
u n + v n s s u n + v n 2 2 ( 2 s ) / ( 2 2 ) u n + v n 2 2 ( s 2 ) / ( 2 2 ) ( γ 2 γ 0 ) 2 ( s 2 ) / ( 2 2 ) ( λ b ) ( 2 s ) / ( 2 2 ) u n + v n λ V s + o ( 1 )
(3.19)
and
z n s s z n 2 2 ( 2 s ) / ( 2 2 ) z n 2 2 ( s 2 ) / ( 2 2 ) 2 ( s 2 ) / 2 ( γ 2 γ 0 ) 2 ( s 2 ) / ( 2 2 ) ( λ b ) ( 2 s ) / ( 2 2 ) z n λ s + o ( 1 ) .
(3.20)
According to (W4), (3.10), (3.17), and (3.18), one gets
λ 2 | z n | R 0 | W u ( x , u n , v n ) + W v ( x , u n , v n ) | | u n + v n | d x λ b 6 | z n | R 0 | z n | | u n + v n | d x λ b 6 z n 2 u n + v n 2 1 6 z n λ u n + v n λ V 1 3 z n + λ 2 + o ( 1 ) .
(3.21)
By virtue of (1.9), (1.10), and (3.9), we have
Φ λ ( u n ) 1 2 Φ λ ( u n ) , u n = λ R N W ˜ ( x , u n , v n ) d x = c λ + o ( 1 ) .
(3.22)
Using (W4), (3.16), (3.19), (3.20) with s = 2 κ / ( κ 1 ) , and (3.22), we obtain
λ 2 | u n | > R 0 | W u ( x , u n , v n ) + W v ( x , u n , v n ) | | u n + v n | d x λ 2 ( | u n | > R 0 ( | W u ( x , u n , v n ) + W v ( x , u n , v n ) | | z n | ) κ d x ) 1 / κ z n s u n + v n s 2 ( s 2 ) / 2 s 2 ( γ 2 γ 0 ) 2 2 ( s 2 ) / s ( 2 2 ) λ ( a 1 | u n | > R 0 W ˜ ( x , u n , v n ) d x ) 1 / κ × ( λ b ) 2 ( 2 s ) / s ( 2 2 ) z n λ u n + v n λ V + o ( 1 ) 1 2 ( 2 a 1 ) 1 / κ ( γ 2 γ 0 ) N / κ λ 1 1 / κ c λ 1 / κ ( λ b ) ( N 2 κ ) / 2 κ z n λ u n + v n λ V + o ( 1 ) ( 2 a 1 ) 1 / κ ( γ 2 γ 0 ) N / κ b ( 2 κ N ) / 2 κ [ λ ( N 2 ) / 2 c λ ] 1 / κ z n + λ 2 + o ( 1 ) 1 3 z n + λ 2 + o ( 1 ) ,
(3.23)
which, together with (1.10), (3.9), (3.21), and (3.23), yields
o ( 1 ) = Φ λ ( z n ) , z n + = z n + λ 2 λ 2 R N [ W u ( x , u n , v n ) + W v ( x , u n , v n ) ] ( u n + v n ) d x 1 3 z n + λ 2 + o ( 1 ) ,
(3.24)
resulting in the fact that z n + λ 0 , which contradicts (3.10). Thus z λ 0 . By a standard argument, we easily verify that Φ λ ( z λ ) = 0 and Φ λ ( z λ ) c λ . Then z λ = ( u λ , v λ ) is a nontrivial solution of (2.1), moreover,
c λ Φ λ ( z λ ) = Φ λ ( z λ ) 1 2 Φ λ ( z λ ) , z λ = λ R N W ˜ ( x , u λ , v λ ) d x .
(3.25)

 □

Theorem 1.1 is a direct consequence of Theorem 1.2.

Declarations

Acknowledgments

This work is partially supported by the NNSF (No. 11471278) of China, Scientific Research Fund of Hunan Provincial Education Department (12C0895), and the Construct Program of the Key Discipline in Hunan Province.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Central South University, Changsha, P.R. China
(2)
Department of Mathematics, Xiangnan University, Chenzhou, P.R. China

References

  1. Clément P, van der Vorst RCAM: On a semilinear elliptic system. Differ. Integral Equ. 1995, 8: 1317-1329.Google Scholar
  2. Clément P, de Figueiredo DG, Mitidieri E: Positive solutions of semilinear elliptic systems. Commun. Partial Differ. Equ. 1992, 17: 923-940. 10.1080/03605309208820869View ArticleGoogle Scholar
  3. de Figueiredo DG, Felmer PL: On superquadratic elliptic systems. Trans. Am. Math. Soc. 1994, 343: 97-116. 10.1090/S0002-9947-1994-1214781-2MathSciNetView ArticleGoogle Scholar
  4. de Figueiredo DG, Ding YH: Strongly indefinite functionals and multiple solutions of elliptic systems. Trans. Am. Math. Soc. 2003, 355: 2973-2989. 10.1090/S0002-9947-03-03257-4MathSciNetView ArticleGoogle Scholar
  5. Hulshof J, van der Vorst RCAM: Differential systems with strongly variational structure. J. Funct. Anal. 1993, 114: 32-58. 10.1006/jfan.1993.1062MathSciNetView ArticleGoogle Scholar
  6. Kryszewski W, Szulkin A: An infinite dimensional Morse theory with applications. Trans. Am. Math. Soc. 1997, 349: 3181-3234. 10.1090/S0002-9947-97-01963-6MathSciNetView ArticleGoogle Scholar
  7. Bartsch T, de Figueiredo DG: Infinitely many solutions of nonlinear elliptic systems. In Topics in Nonlinear Analysis. Birkhäuser, Basel; 1999:51-67. 10.1007/978-3-0348-8765-6_4View ArticleGoogle Scholar
  8. de Figueiredo DG, Yang J: Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal. 1998, 33: 211-234. 10.1016/S0362-546X(97)00548-8MathSciNetView ArticleGoogle Scholar
  9. Li G, Yang J: Asymptotically linear elliptic systems. Commun. Partial Differ. Equ. 2004, 29: 925-954. 10.1081/PDE-120037337View ArticleGoogle Scholar
  10. Ávila AI, Yang J: Multiple solutions of nonlinear elliptic systems. NoDEA Nonlinear Differ. Equ. Appl. 2005, 12: 459-479. 10.1007/s00030-005-0022-7View ArticleGoogle Scholar
  11. Ávila AI, Yang J: On the existence and shape of least energy solutions for some elliptic systems. J. Differ. Equ. 2003, 191: 348-376. 10.1016/S0022-0396(03)00017-2View ArticleGoogle Scholar
  12. Bartsch T, Ding YH: Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math. Nachr. 2006, 279: 1267-1288. 10.1002/mana.200410420MathSciNetView ArticleGoogle Scholar
  13. Li GB, Szulkin A: An asymptotically periodic Schrödinger equation with indefinite linear part. Commun. Contemp. Math. 2002, 4: 763-776. 10.1142/S0219199702000853MathSciNetView ArticleGoogle Scholar
  14. Kryszewki W, Szulkin A: Generalized linking theorem with an application to semilinear Schrödinger equation. Adv. Differ. Equ. 1998, 3: 441-472.Google Scholar
  15. Sirakov B:On the existence of solutions of Hamiltonian elliptic systems in R N . Adv. Differ. Equ. 2000, 5: 1445-1464.MathSciNetGoogle Scholar
  16. Zhang J, Tang XH, Zhang W: Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms. Nonlinear Anal. 2014, 95: 1-10. 10.1016/j.na.2013.07.027MathSciNetView ArticleGoogle Scholar
  17. Zhang J, Qin WP, Zhao FK: Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system. J. Math. Anal. Appl. 2013, 399: 433-441. 10.1016/j.jmaa.2012.10.030MathSciNetView ArticleGoogle Scholar
  18. Zhao F, Zhao L, Ding Y: Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems. ESAIM Control Optim. Calc. Var. 2010, 16: 77-91. 10.1051/cocv:2008064MathSciNetView ArticleGoogle Scholar
  19. Zhao F, Zhao L, Ding Y:Multiple solution for a superlinear and periodic elliptic system on R N . Z. Angew. Math. Phys. 2011, 62: 495-511. 10.1007/s00033-010-0105-0MathSciNetView ArticleGoogle Scholar
  20. Ambrosetti A, Badiale M, Cingolani S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 1997, 140: 285-300. 10.1007/s002050050067MathSciNetView ArticleGoogle Scholar
  21. Floer A, Weinstein A: Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential. J. Funct. Anal. 1986, 69: 397-408. 10.1016/0022-1236(86)90096-0MathSciNetView ArticleGoogle Scholar
  22. Oh YG:Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class ( V ) α . Commun. Partial Differ. Equ. 1988, 13: 1499-1519. 10.1080/03605308808820585View ArticleGoogle Scholar
  23. Oh YG: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 1990, 131: 223-253. 10.1007/BF02161413View ArticleGoogle Scholar
  24. del Pino M, Felmer P: Multipeak bound states of nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1998, 15: 127-149. 10.1016/S0294-1449(97)89296-7MathSciNetView ArticleGoogle Scholar
  25. del Pino M, Felmer P: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 2002, 324: 1-32. 10.1007/s002080200327MathSciNetView ArticleGoogle Scholar
  26. Rabinowitz PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 1992, 43: 270-291. 10.1007/BF00946631MathSciNetView ArticleGoogle Scholar
  27. Zhang J, Zhao FK: Multiple solutions for a semiclassical Schrödinger equation. Nonlinear Anal. 2012, 75: 1834-1842. 10.1016/j.na.2011.09.032MathSciNetView ArticleGoogle Scholar
  28. Lin X, Tang XH: Semiclassical solutions of perturbed p -Laplacian equations with critical nonlinearity. J. Math. Anal. Appl. 2014, 413: 439-449. 10.1016/j.jmaa.2013.11.063MathSciNetView ArticleGoogle Scholar
  29. Ding YH, Lee C, Zhao FK: Semiclassical limits of ground state solutions to Schrödinger systems. Calc. Var. 2013.Google Scholar
  30. Ramos M: On singular perturbations of superlinear elliptic systems. J. Math. Anal. Appl. 2009, 352: 246-258. 10.1016/j.jmaa.2008.06.019MathSciNetView ArticleGoogle Scholar
  31. Ramos M, Tavares H: Solutions with multiple spike patterns for an elliptic system. Calc. Var. 2008, 31: 1-25. 10.1007/s00526-007-0103-zMathSciNetView ArticleGoogle Scholar
  32. Pistoia A, Ramos M: Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions. J. Differ. Equ. 2004, 201: 160-176. 10.1016/j.jde.2004.02.003MathSciNetView ArticleGoogle Scholar
  33. Sirakov B, Soares SHM: Soliton solutions to systems of coupled Schrödinger equations of Hamiltonian type. Trans. Am. Math. Soc. 2010, 362: 5729-5744. 10.1090/S0002-9947-2010-04982-7MathSciNetView ArticleGoogle Scholar
  34. Xiao L, Wang J, Fan M, Zhang F: Existence and multiplicity of semiclassical solutions for asymptotically Hamiltonian elliptic systems. J. Math. Anal. Appl. 2013, 399: 340-351. 10.1016/j.jmaa.2012.10.010MathSciNetView ArticleGoogle Scholar
  35. Zhang J, Tang XH, Zhang W: Semiclassical solutions for a class of Schrödinger system with magnetic potentials. J. Math. Anal. Appl. 2014, 414: 357-371. 10.1016/j.jmaa.2013.12.060MathSciNetView ArticleGoogle Scholar
  36. Ding YH, Lee C: Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J. Differ. Equ. 2006, 222: 137-163. 10.1016/j.jde.2005.03.011MathSciNetView ArticleGoogle Scholar
  37. Ding YH: Variational Methods for Strongly Indefinite Problems. World Scientific, Hackensack; 2008.Google Scholar
  38. Ding YH, Lee C: Periodic solutions of an infinite dimensional Hamiltonian system. Rocky Mt. J. Math. 2005, 35: 1881-1908. 10.1216/rmjm/1181069621MathSciNetView ArticleGoogle Scholar
  39. Qin DD, Tang XH, Jian Z: Multiple solutions for semilinear elliptic equations with sign-changing potential and nonlinearity. Electron. J. Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-305Google Scholar
  40. Tang XH: Infinitely many solutions for semilinear Schrödinger equation with sign-changing potential and nonlinearity. J. Math. Anal. Appl. 2013, 401: 407-415. 10.1016/j.jmaa.2012.12.035MathSciNetView ArticleGoogle Scholar
  41. Tang XH: New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation. Adv. Nonlinear Stud. 2014, 14: 349-361.Google Scholar
  42. Tang XH: New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum. J. Math. Anal. Appl. 2014, 413: 392-410. 10.1016/j.jmaa.2013.11.062MathSciNetView ArticleGoogle Scholar
  43. Tang XH: Non-Nehari manifold method for superlinear Schrödinger equation. Taiwan. J. Math. 2014.Google Scholar
  44. Zhang RM, Chen J, Zhao FK: Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete Contin. Dyn. Syst., Ser. A 2011, 30: 1249-1262. 10.3934/dcds.2011.30.1237MathSciNetView ArticleGoogle Scholar
  45. Liao FF, Tang XH, Zhang J: Existence of solutions for periodic elliptic system with general superlinear nonlinearity. Z. Angew. Math. Phys. 2014.Google Scholar
  46. Sirakov B:Standing wave solutions of the nonlinear Schrödinger equations in R N . Ann. Mat. 2002, 183: 73-83. 10.1007/s102310200029MathSciNetView ArticleGoogle Scholar
  47. Ding YH, Wei JC: Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials. J. Funct. Anal. 2007, 251: 546-572. 10.1016/j.jfa.2007.07.005MathSciNetView ArticleGoogle Scholar
  48. Lions PL: The concentration-compactness principle in the calculus of variations. The locally compact case, part 2. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1984, 1: 223-283.Google Scholar
  49. Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.View ArticleGoogle Scholar

Copyright

© Liao et al.; licensee Springer 2014

This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Advertisement