Open Access

Existence of positive ground states for some nonlinear Schrödinger systems

Boundary Value Problems20132013:13

DOI: 10.1186/1687-2770-2013-13

Received: 26 March 2012

Accepted: 12 January 2013

Published: 28 January 2013

Abstract

We prove the existence of positive ground states for the nonlinear Schrödinger system

{ Δ u + ( 1 + a ( x ) ) u = F u ( u , v ) + λ v , Δ v + ( 1 + b ( x ) ) v = F v ( u , v ) + λ u , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equa_HTML.gif

where a, b are periodic or asymptotically periodic and F satisfies some superlinear conditions in ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq1_HTML.gif. The proof is based on the method of Nehari manifold and the concentration-compactness principle.

MSC:35J05, 35J50, 35J61.

Keywords

nonlinear Schrödinger system Nehari manifold lack of compactness ground state

1 Introduction and statement of the main result

This paper was motivated by the following two-component system of nonlinear Schrödinger equations:
{ i t ϕ = Δ ϕ + μ 1 | ϕ | 2 ϕ + β | φ | 2 ϕ , x R n , t > 0 , i t φ = Δ φ + μ 2 | φ | 2 φ + β | ϕ | 2 φ , x R n , t > 0 , ϕ = ϕ ( x , t ) , φ = φ ( x , t ) C , ϕ ( x , t ) , φ ( x , t ) 0 as  | x | , t > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ1_HTML.gif
(1.1)
where μ i > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq2_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq3_HTML.gif, β R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq4_HTML.gif and n = 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq5_HTML.gif. The system (1.1) has applications in many physical problems, especially in nonlinear optics (see [1]). To obtain standing wave solutions of (1.1) of the form ϕ ( x , t ) = e i λ 1 t u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq6_HTML.gif, φ ( x , t ) = e i λ 2 t v ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq7_HTML.gif with λ 1 , λ 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq8_HTML.gif, the system (1.1) turns out to be
{ Δ u + λ 1 u = μ 1 u 3 + β u v 2 , Δ v + λ 2 v = μ 2 v 3 + β v u 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ2_HTML.gif
(1.2)

Following the work [2] by Lin and Wei about the existence of ground states for the problem (1.2), there are many results on the existence of ground states relevant to five parameters ( λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq9_HTML.gif, λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq10_HTML.gif, μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq11_HTML.gif, μ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq12_HTML.gif and β); see [39] and the references therein. Later in [10], assuming λ 1 = λ 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq13_HTML.gif, Pomponio and Secchi established the existence of radially symmetric ground states for (1.2) with general nonlinearities ( f ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq14_HTML.gif and g ( v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq15_HTML.gif).

On the other hand, some authors considered the existence of ground states for non-autonomous similar problems. We recall the results about non-autonomous case for two subcases. For periodic case, in [11] Szulkin and Weth referred that treating as periodic Schrödinger equations, it is possible to deduce that there are ground states for the following system using the method of Nehari manifold:
{ Δ u + u = G u ( x , u , v ) , Δ v + v = G v ( x , u , v ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ3_HTML.gif
(1.3)
where G is periodic in x and satisfies some superlinear conditions in ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq1_HTML.gif. For non-periodic case, we refer to [8, 1214] for instance. As we can observe, most of the previous results on ground states for the non-periodic system have used the condition that there exists a limit system (or the problem at infinity; for precise statement, refer to [15]). Moreover, the limit system is autonomous. Here we mainly deal with an asymptotically periodic Schrödinger system which has a periodic non-autonomous limit system, roughly speaking. In this paper, we are concerned with the existence of positive ground states for the nonlinear Schrödinger system in W 1 , 2 ( R N ) × W 1 , 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq16_HTML.gif ( N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq17_HTML.gif)
{ Δ u + ( 1 + a ( x ) ) u = F u ( u , v ) + λ v , Δ v + ( 1 + b ( x ) ) v = F v ( u , v ) + λ u , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ4_HTML.gif
(NLS)
where λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq18_HTML.gif is a real parameter. For simplicity, we denote +∞ by ∞
2 = { 2 N N 2 , N 3 , , N = 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equb_HTML.gif

Moreover, in what follows, the notation inf (sup) is understood as the essential infimum (supremum). In the sequel, let a , b L ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq19_HTML.gif and F C 1 ( R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq20_HTML.gif with F ( 0 , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq21_HTML.gif, we always assume that

(V1) inf R N { 1 + a ( x ) } > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq22_HTML.gif, inf R N { 1 + b ( x ) } > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq23_HTML.gif,

(F1) | F ( u , v ) | C 0 ( 1 + | ( u , v ) | q 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq24_HTML.gif for some C 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq25_HTML.gif and 2 < q < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq26_HTML.gif,

(F2) | F ( u , v ) | = o ( | ( u , v ) | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq27_HTML.gif as | ( u , v ) | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq28_HTML.gif,

(F3) ( u , v ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq29_HTML.gif, s > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq30_HTML.gif, s F ( s u , s v ) ( u , v ) s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq31_HTML.gif is strictly increasing,

(F4) F ( u , v ) u 2 + v 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq32_HTML.gif as | ( u , v ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq33_HTML.gif,

(F5) F u ( u , v ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq34_HTML.gif, F v ( u , v ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq35_HTML.gif, F u ( 0 , v ) = F v ( u , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq36_HTML.gif, u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq37_HTML.gif, v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq38_HTML.gif,

(F6) F ( u , v ) F ( | u | , | v | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq39_HTML.gif, u , v R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq40_HTML.gif.

(F1)-(F4) are similar to the conditions of the nonlinearities for the periodic system (1.3) as considered in [11]. We divide the study of (NLS) into two cases as follows.

First, we consider the periodic case

(V2) a ( x ) = a ( x + y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq41_HTML.gif, b ( x ) = b ( x + y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq42_HTML.gif, x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq43_HTML.gif, y Z N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq44_HTML.gif.

We have the following result.

Theorem 1.1 Let (V1), (V2) and (F1)-(F6) hold. Then the system (NLS) has a positive ground state.

Remark 1.1 It is observed that the system (NLS) with periodic a and b is a particular case of the problem (1.3) with
G ( x , u , v ) = F ( u , v ) + λ u v 1 2 ( a ( x ) u 2 + b ( x ) v 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equc_HTML.gif

and G is periodic in x. The problem (1.3) is mentioned in [11] when G is periodic in x. However, in [11] the conditions on the function G are not made explicit.

Next, we consider the asymptotically periodic case. We assume that there are functions a p , b p L ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq45_HTML.gif satisfying (V1) and (V2) and a, b satisfies that

(V3) lim | x | | a ( x ) a p ( x ) | = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq46_HTML.gif, lim | x | | b ( x ) b p ( x ) | = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq47_HTML.gif,

(V4) a a p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq48_HTML.gif, b b p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq49_HTML.gif.

We have the following result.

Theorem 1.2 Assume that a p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq50_HTML.gif and b p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq51_HTML.gif satisfy (V2). Let (V1), (V3), (V4) and (F1)-(F6) hold. Then the system (NLS) has a positive ground state.

Remark 1.2 Conditions (V1) and (V4) imply that a p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq50_HTML.gif and b p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq51_HTML.gif satisfy (V1).

In addition, we consider the following conditions:

(V5) a p = b p : = V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq52_HTML.gif, a + b 2 V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq53_HTML.gif,

(F7) F u ( u , v ) = F v ( v , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq54_HTML.gif, u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq55_HTML.gif, v > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq56_HTML.gif.

We have the following result.

Theorem 1.3 Suppose that a p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq50_HTML.gif and b p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq51_HTML.gif satisfy (V1) and (V2). Let (V1), (V3), (V5) and (F1)-(F7) hold. Then the system (NLS) has a positive ground state.

We will prove Theorems 1.1, 1.2 and 1.3 using the method of Nehari manifold. We first reduce the problem of seeking for ground states of (NLS) into that of looking for minimizers of the functional constrained on the Nehari manifold. Then we apply the concentration-compactness principle to solve the minimization problem. Since the Nehari manifold for (NLS) may not be smooth, in the same way as [11], we will make use of the differential structure of a unit sphere in W 1 , 2 ( R N ) × W 1 , 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq16_HTML.gif to find a ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq57_HTML.gif sequence (c is the infimum of the functional constrained on the Nehari manifold). When (NLS) is periodic, we will use the invariance of the functional under translation to recover the compactness of the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq57_HTML.gif sequence. When the system (NLS) is asymptotically periodic, the difficulty is to recover the compactness for the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq57_HTML.gif sequence. By comparing c with the infimum of the functional of the related periodic limit system constrained on the corresponding Nehari manifold, we will restore the compactness.

The paper is organized as follows. In Section 2 we give some preliminaries. In Section 3 we introduce the variational setting. In Section 4 we consider the periodic case and prove Theorem 1.1. Section 5 is devoted to studying the asymptotically periodic case and showing Theorems 1.2 and 1.3.

2 Notation and preliminaries

We use the following notation:

  • For simplicity, we denote h : = R N h ( x ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq58_HTML.gif and E h : = E h ( x ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq59_HTML.gif, where E R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq60_HTML.gif is measurable.

  • X denotes the Sobolev space W 1 , 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq61_HTML.gif ( N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq17_HTML.gif), with the standard scalar product u , v X = ( u v + u v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq62_HTML.gif and the norm u X 2 = u , u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq63_HTML.gif. H = X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq64_HTML.gif with the norm ( u , v ) H 2 = u X 2 + v X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq65_HTML.gif. When there is no possible misunderstanding, the subscripts could be omitted.

  • The usual norm in L r ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq66_HTML.gif ( 2 r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq67_HTML.gif) will be denoted by | | r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq68_HTML.gif.

  • S = { ( u , v ) H : ( u , v ) 2 = 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq69_HTML.gif.

  • For any ϱ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq70_HTML.gif and z R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq71_HTML.gif, B ϱ ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq72_HTML.gif denotes the ball of radius ϱ centered at z.

Note that a , b L ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq73_HTML.gif and F C 1 ( R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq20_HTML.gif. Then by conditions (F1) and (F2), the functional
Φ ( u , v ) = 1 2 ( u 2 + v 2 + a ( x ) u 2 + b ( x ) v 2 ) λ u v F ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equd_HTML.gif
is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq74_HTML.gif and its critical points are solutions of (NLS). Moreover, by (V1) we have
μ ( u , v ) 2 u 2 + v 2 + a ( x ) u 2 + b ( x ) v 2 2 λ u v ν ( u , v ) 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ5_HTML.gif
(2.1)

with μ , ν > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq75_HTML.gif.

A solution ( u ˜ , v ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq76_HTML.gif of (NLS) is called a ground state if
Φ ( u ˜ , v ˜ ) = min { Φ ( u , v ) : ( u , v ) H { ( 0 , 0 ) } , Φ ( u , v ) = 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Eque_HTML.gif

A ground state ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq1_HTML.gif such that u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq55_HTML.gif, v > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq56_HTML.gif ( u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq37_HTML.gif, v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq38_HTML.gif) is called a positive (non-negative) ground state. Below we give some lemmas useful for studying our problem.

Lemma 2.1 (F1) and (F2) imply that for all ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq77_HTML.gif, there exists C ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq78_HTML.gif such that
| F ( u , v ) | ϵ | ( u , v ) | + C ϵ | ( u , v ) | q 1 u , v R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ6_HTML.gif
(2.2)
(F2) and (F3) yield that
F ( u , v ) > 0 , 2 F ( u , v ) < F ( u , v ) ( u , v ) ( u , v ) ( 0 , 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ7_HTML.gif
(2.3)

Moreover, (F3) implies the function α ( s ) = 1 2 F ( s u , s v ) ( s u , s v ) F ( s u , s v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq79_HTML.gif is increasing in ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq80_HTML.gif for all u , v R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq81_HTML.gif.

Proof The inequalities (2.2) and (2.3) are easily inferred from the corresponding assumptions. We just prove the last conclusion. Indeed, let s 2 s 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq82_HTML.gif. Then by (F3) we obtain
α ( s 2 ) α ( s 1 ) = 1 2 F ( s 2 u , s 2 v ) ( s 2 u , s 2 v ) 1 2 F ( s 1 u , s 1 v ) ( s 1 u , s 1 v ) + F ( s 1 u , s 1 v ) F ( s 2 u , s 2 v ) = 0 s 2 t F ( s 2 u , s 2 v ) ( u , v ) s 2 d t 0 s 1 t F ( s 1 u , s 1 v ) ( u , v ) s 1 d t s 1 s 2 F ( t u , t v ) ( u , v ) d t = 0 s 1 t [ F ( s 2 u , s 2 v ) ( u , v ) s 2 F ( s 1 u , s 1 v ) ( u , v ) s 1 ] d t + s 1 s 2 t [ F ( s 2 u , s 2 v ) ( u , v ) s 2 F ( t u , t v ) ( u , v ) t ] d t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equf_HTML.gif

 □

Lemma 2.2 Let (F1) and (F2) hold. Then Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq83_HTML.gif is weakly sequentially continuous. Namely, if ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq84_HTML.gif in H, then Φ ( u n , v n ) Φ ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq85_HTML.gif in H.

Proof Suppose ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq84_HTML.gif in H. After passing to a subsequence, we assume ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq86_HTML.gif in L loc q ( R N ) × L loc q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq87_HTML.gif. By (F1), we get
F u ( u n , v n ) F u ( u , v ) , F v ( u n , v n ) F v ( u , v ) in  L loc q q 1 ( R N ) × L loc q q 1 ( R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equg_HTML.gif
Then for all ( ϕ , φ ) C 0 ( R N ) × C 0 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq88_HTML.gif, we have
F u ( u n , v n ) ϕ F u ( u , v ) ϕ , F v ( u n , v n ) φ F v ( u , v ) φ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equh_HTML.gif
So, one easily has that
Φ ( u n , v n ) , ( ϕ , φ ) Φ ( u , v ) , ( ϕ , φ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ8_HTML.gif
(2.4)
Now, we claim that Φ ( u n , v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq89_HTML.gif is bounded in H. Indeed, for ( h , k ) H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq90_HTML.gif, using (2.2) and the Hölder inequality, we obtain that
| F u ( u n , v n ) h | C ( | u n | + | v n | ) | h | + C ¯ ( | u n | q 1 + | v n | q 1 ) | h | C ( | u n | 2 + | v n | 2 ) | h | 2 + C 1 C ¯ ( | u n | q q 1 + | v n | q q 1 ) | h | q C ( u n + v n ) h + C 2 C ¯ ( u n q 1 + v n q 1 ) h < C h . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equi_HTML.gif
Similarly, we get | F v ( u n , v n ) k | C k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq91_HTML.gif. Then we easily have
| Φ ( u n , v n ) , ( h , k ) | C ( h , k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equj_HTML.gif

Hence, Φ ( u n , v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq89_HTML.gif is bounded in H. Combining with the fact that C 0 ( R N ) × C 0 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq92_HTML.gif is dense in H, we easily deduce that (2.4) holds for any ( ϕ , φ ) H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq93_HTML.gif. Therefore, Φ ( u n , v n ) Φ ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq85_HTML.gif in H. □

3 Variational setting

This section is devoted to describing the variational framework for the study of ground states for (NLS).

It is easy to see that Φ is bounded neither from above nor from below. So, it is convenient to consider Φ on the Nehari manifold that contains all nontrivial critical points of Φ and on which Φ turns out to be bounded from below. The Nehari manifold M corresponding to Φ is defined by
M = { ( u , v ) H { ( 0 , 0 ) } : Φ ( u , v ) , ( u , v ) = 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equk_HTML.gif
where
Φ ( u , v ) , ( u , v ) = u 2 + u 2 + a ( x ) u 2 + b ( x ) v 2 2 λ u v F ( u , v ) ( u , v ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equl_HTML.gif

Below we investigate the main properties of Φ on M.

Lemma 3.1 Let (F2) and (F3) hold. Then Φ is bounded from below on M by 0.

Proof

Note that
Φ | M ( u , v ) = [ 1 2 F ( u , v ) ( u , v ) F ( u , v ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ9_HTML.gif
(3.1)

By (2.3) we have Φ | M ( u , v ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq94_HTML.gif. □

Define the least energy of (NLS) on M by c : = inf Φ | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq95_HTML.gif, then c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq96_HTML.gif. Next, we prove M is a manifold. First, we give the following two lemmas, which will be important when proving M is a manifold.

Lemma 3.2 Let (V1) and (F2)-(F4) hold. Assume ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq84_HTML.gif in H and ( u , v ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq97_HTML.gif. Then for any { t n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq98_HTML.gif with t n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq99_HTML.gif and t n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq100_HTML.gif, we have
F ( t n u n , t n v n ) t n 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equm_HTML.gif

Moreover, Φ ( t n u n , t n v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq101_HTML.gif.

Proof Since ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq84_HTML.gif in H, we assume that ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq86_HTML.gif in L loc 2 ( R N ) × L loc 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq102_HTML.gif, and ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq86_HTML.gif a.e. on R 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq103_HTML.gif for a subsequence. By ( u , v ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq97_HTML.gif, there exists a positive measure set Ω such that ( u ( x ) , v ( x ) ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq104_HTML.gif, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq105_HTML.gif. Then t n | ( u n ( x ) , v n ( x ) ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq106_HTML.gif, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq107_HTML.gif. By (F4) we have
Ω lim ̲ F ( t n u n , t n v n ) t n 2 ( u n 2 + v n 2 ) ( u n 2 + v n 2 ) = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equn_HTML.gif
Therefore, (2.3) and the Fatou lemma yield that
lim ̲ F ( t n u n , t n v n ) t n 2 = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equo_HTML.gif
Using (2.1) we have
Φ ( t n ( u n , v n ) ) t n 2 2 [ ν ( u n 2 + v n 2 ) 2 F ( t n u n , t n v n ) t n 2 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equp_HTML.gif

since { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq108_HTML.gif is bounded in H. □

Lemma 3.3 Let (V1) and (F1)-(F4) hold. Then
  1. (i)

    for each ( u , v ) H { ( 0 , 0 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq109_HTML.gif, there exists t ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq110_HTML.gif such that if g ( u , v ) ( t ) : = Φ ( t u , t v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq111_HTML.gif, then g ( u , v ) ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq112_HTML.gif for 0 < t < t ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq113_HTML.gif and g ( u , v ) ( t ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq114_HTML.gif for t > t ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq115_HTML.gif;

     
  2. (ii)

    there exists ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq116_HTML.gif such that t ( w , z ) ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq117_HTML.gif for all ( w , z ) S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq118_HTML.gif;

     
  3. (iii)

    for each compact subset W S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq119_HTML.gif, there exists a constant C W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq120_HTML.gif such that t ( u , v ) C W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq121_HTML.gif for all ( u , v ) W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq122_HTML.gif.

     
Proof
  1. (i)
    Note that
    g ( u , v ) ( t ) = t [ u 2 + v 2 + a ( x ) u 2 + b ( x ) v 2 2 λ u v F ( t u , t v ) ( u , v ) t ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equq_HTML.gif
     
Using (F2), we infer that when t is small enough, g ( u , v ) ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq112_HTML.gif. By Lemma 3.2 and (2.3), we have
F ( t u , t v ) ( u , v ) t 2 F ( t u , t v ) t 2 , t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equr_HTML.gif
Then when t is large enough, g ( u , v ) ( t ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq114_HTML.gif. Then g ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq123_HTML.gif has maximum points in ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq80_HTML.gif. Moreover, from (F3) one easily deduces that the critical point of g ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq123_HTML.gif is unique in ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq80_HTML.gif, and then it is the maximum point. We denote it by t ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq110_HTML.gif. Then g ( u , v ) ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq112_HTML.gif for 0 < t < t ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq113_HTML.gif and g ( u , v ) ( t ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq114_HTML.gif for t > t ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq115_HTML.gif.
  1. (ii)
    If ( u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq124_HTML.gif, then
    u 2 + v 2 + a ( x ) u 2 + b ( x ) v 2 2 λ u v = F ( u , v ) ( u , v ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equs_HTML.gif
     
By (2.1) and (2.2), we get
μ ( u 2 + v 2 ) ϵ ( | u | 2 2 + | v | 2 2 ) + C ϵ ( | u | q q + | v | q q ) ϵ ( u 2 + v 2 ) + C C ϵ ( u q + v q ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equt_HTML.gif
where ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq77_HTML.gif is arbitrary. Then
u q 2 + v q 2 C ˜ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equu_HTML.gif
So, there exists ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq116_HTML.gif such that
u 2 + v 2 ρ 2 for all  ( u , v ) M . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ10_HTML.gif
(3.2)
Using (i), for ( w , z ) S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq118_HTML.gif, there exists t ( w , z ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq125_HTML.gif such that g ( w , z ) ( t ( w , z ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq126_HTML.gif. Then t ( w , z ) ( w , z ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq127_HTML.gif. Then (3.2) yields the conclusion (ii).
  1. (iii)

    We argue by contradiction. Suppose that there exist a compact set W and a sequence { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq108_HTML.gif such that { ( u n , v n ) } W S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq128_HTML.gif and t ( u n , v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq129_HTML.gif. Since W is compact, there exists ( u , v ) W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq122_HTML.gif such that ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq86_HTML.gif in H. Then Lemma 3.2 implies that Φ ( t ( u n , v n ) ( u n , v n ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq130_HTML.gif. Contrary to Lemma 3.1 since t ( u n , v n ) ( u n , v n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq131_HTML.gif. This ends the proof. □

     
Remark 3.1 Lemma 3.3(i) implies that for each ( u , v ) H { ( 0 , 0 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq109_HTML.gif, there exists a unique t ( u , v ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq132_HTML.gif such that
t ( u , v ) ( u , v ) M and Φ ( t ( u , v ) ( u , v ) ) = max t > 0 Φ ( t u , t v ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ11_HTML.gif
(3.3)

As a consequence of Lemma 3.3(i), we can define the mapping m : S M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq133_HTML.gif by m ( u , v ) = t ( u , v ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq134_HTML.gif. By Lemma 3.3, [[11], Proposition 3.1(b)] yields the following result.

Lemma 3.4 If (V1) and (F1)-(F4) are satisfied, then m is a homeomorphism between S and M, and M is a manifold.

If M is a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq74_HTML.gif manifold, we can make use of the differential structure of M to reduce the problem of finding a ground state for (NLS) into that of looking for a minimizer of Φ | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq135_HTML.gif and solve the minimizing problem. However, since F C 1 ( R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq20_HTML.gif, M may not be a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq74_HTML.gif manifold. Noting that M and S are homeomorphic, we will take advantage of the differential structure of S to seek for ground states for (NLS) as [11]. Therefore, as in [11], we introduce the functional Ψ : S R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq136_HTML.gif defined by Ψ ( u , v ) : = Φ ( m ( u , v ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq137_HTML.gif, and we have the following conclusion.

Proposition 3.1 Let (V1) and (F1)-(F4) hold. Then the following results hold:
  1. (i)

    If { ( w n , z n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq138_HTML.gif is a PS sequence for Ψ, then { m ( w n , z n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq139_HTML.gif is a PS sequence for Φ.

     
  2. (ii)

    ( w , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq140_HTML.gif is a critical point of Ψ if and only if m ( w , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq141_HTML.gif is a nontrivial critical point of Φ. Moreover, inf S Ψ = inf M Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq142_HTML.gif.

     
  3. (iii)

    A minimizer of Φ | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq135_HTML.gif is a solution of (NLS).

     

Proof As in the proof of [[11], Corollary 3.3], we can show (i) and (ii). Now, we prove the conclusion (iii). Indeed, let ( u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq124_HTML.gif such that Φ ( u , v ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq143_HTML.gif. Then Ψ ( w , z ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq144_HTML.gif, where ( w , z ) = m 1 ( u , v ) S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq145_HTML.gif. By the conclusion (ii), we have Ψ ( w , z ) = inf S Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq146_HTML.gif. So, Ψ ( w , z ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq147_HTML.gif. Using the conclusion (ii) again, we deduce that Φ ( u , v ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq148_HTML.gif. □

From the definition of a ground state, we translate the problem of looking for a ground state for (NLS) into that of seeking for a solution for (NLS) which is a minimizer of Φ | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq135_HTML.gif. By Proposition 3.1(iii), in order to look for a ground state for (NLS), we just need to seek for a minimizer of Φ | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq135_HTML.gif.

4 The periodic case

In this section, we consider the periodic case and prove Theorem 1.1. In [11], Szulkin and Weth considered the existence of ground states for periodic single Schrödinger equations. Treating as in [11], we find ground states for a periodic case for the system (NLS). In addition, under conditions (F5) and (F6), we deduce that there are positive ground states.

From the statement in Section 3, it suffices to solve the minimizing problem. By conclusions (i) and (ii) of Proposition 3.1, we first make use of the minimizing sequence of Ψ to obtain a ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq57_HTML.gif sequence of Φ. Then we use the invariant of the functional under translation of the form v v ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq149_HTML.gif, y Z N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq150_HTML.gif to recover the compactness for the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq57_HTML.gif sequence.

Proof of Theorem 1.1 Let ( w ¯ n , z ¯ n ) S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq151_HTML.gif be a minimizing sequence of Ψ. By the Ekeland variational principle [[16], Theorem 8.5], we may assume that Ψ ( w ¯ n , z ¯ n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq152_HTML.gif. Using Proposition 3.1(i), we have that Φ ( u n , v n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq153_HTML.gif, where ( u n , v n ) = m ( w ¯ n , z ¯ n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq154_HTML.gif. Proposition 3.1(ii) implies that Φ ( u n , v n ) = Ψ ( w ¯ n , z ¯ n ) inf S Ψ = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq155_HTML.gif.

We claim that { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq108_HTML.gif is bounded in H. Otherwise, suppose s n : = ( u n , v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq156_HTML.gif up to a subsequence. Set ( w n , z n ) = ( u n , v n ) ( u n , v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq157_HTML.gif. Then we assume ( w n , z n ) ( w , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq158_HTML.gif in H, ( w n , z n ) ( w , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq159_HTML.gif in L loc 2 ( R N ) × L loc 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq102_HTML.gif and ( w n , z n ) ( w , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq159_HTML.gif a.e. on R 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq103_HTML.gif after passing to a subsequence. Moreover, the Sobolev embedding theorem implies that { ( w n , z n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq138_HTML.gif is bounded in L q ( R N ) × L q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq160_HTML.gif, namely, { | w n | q + | z n | q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq161_HTML.gif is bounded. Taking a subsequence, we suppose | w n | q + | z n | q A [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq162_HTML.gif.
  1. (i)
    If A = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq163_HTML.gif, then for any ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq77_HTML.gif, there exists K N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq164_HTML.gif such that | w n | q q + | z n | q q < C ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq165_HTML.gif, for n > K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq166_HTML.gif. Combining with (2.2), for n > K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq166_HTML.gif and s > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq30_HTML.gif, we have
    F ( s w n , s z n ) ϵ s 2 ( | w n | 2 2 + | z n | 2 2 ) + s q C C ϵ ( | w n | q q + | z n | q q ) < C ϵ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equv_HTML.gif
     
Then F ( s w n , s z n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq167_HTML.gif. Hence, by (2.1) we get
c + o n ( 1 ) = Φ ( u n , v n ) Φ ( s w n , s z n ) s 2 μ 2 ( w n 2 + z n 2 ) F ( s w n , s z n ) s 2 μ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equw_HTML.gif
a contradiction for s > 2 c μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq168_HTML.gif.
  1. (ii)
    If A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq169_HTML.gif, then we can assume that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq170_HTML.gif in L q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq171_HTML.gif. From the Lions compactness lemma [[16], Lemma 1.21], it follows that there exist δ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq172_HTML.gif and x n R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq173_HTML.gif such that
    B 1 ( x n ) | w n | 2 > δ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ12_HTML.gif
    (4.1)
     

Since Φ and M are invariant by translation of the form v v ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq149_HTML.gif, y Z N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq44_HTML.gif, translating w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq174_HTML.gif if necessary, we may assume { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq175_HTML.gif is bounded. Since w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq176_HTML.gif in L loc 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq177_HTML.gif, then (4.1) implies w 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq178_HTML.gif. Then from Lemma 3.2, we deduce that Φ ( s n w n , s n z n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq179_HTML.gif. This is impossible since Φ ( s n w n , s n z n ) = Φ ( u n , v n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq180_HTML.gif.

Hence, { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq108_HTML.gif is bounded in H. Suppose that ( u n , v n ) ( u ˇ , v ˇ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq181_HTML.gif in H, ( u n , v n ) ( u ˇ , v ˇ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq182_HTML.gif in L loc 2 ( R N ) × L loc 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq102_HTML.gif and ( u n , v n ) ( u ˇ , v ˇ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq182_HTML.gif a.e. on R 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq103_HTML.gif for a subsequence. Since Φ ( u n , v n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq153_HTML.gif, Lemma 2.2 yields Φ ( u ˇ , v ˇ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq183_HTML.gif.

We will show that ( u ˇ , v ˇ ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq184_HTML.gif. Similarly, suppose | u n | q + | v n | q B [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq185_HTML.gif. If B = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq186_HTML.gif, then as before, combining with (2.2), we obtain that F ( u n , v n ) ( u n , v n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq187_HTML.gif. Hence, by (2.1) we have
o n ( 1 ) = Φ ( u n , v n ) , ( u n , v n ) μ ( u n 2 + v n 2 ) F ( u n , v n ) ( u n , v n ) = μ ( u n 2 + v n 2 ) + o n ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equx_HTML.gif
Then ( u n , v n ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq188_HTML.gif in H. This is impossible since ( u n , v n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq189_HTML.gif and (3.2) holds. Therefore, B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq190_HTML.gif. So, we can assume https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq191_HTML.gif in L q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq171_HTML.gif. Then the Lions compactness lemma implies that there exist y n R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq192_HTML.gif, δ ˜ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq193_HTML.gif such that
B 1 ( y n ) | u n | 2 > δ ˜ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ13_HTML.gif
(4.2)
As before, translating u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq194_HTML.gif if necessary, we may assume { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq195_HTML.gif is bounded. Since (4.2) and u n u ˇ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq196_HTML.gif in L loc 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq177_HTML.gif, we get u ˇ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq197_HTML.gif. Note that Φ ( u ˇ , v ˇ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq183_HTML.gif. So, ( u ˇ , v ˇ ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq198_HTML.gif. Then by (3.1) we get
c + o n ( 1 ) = Φ ( u n , v n ) = [ 1 2 F ( u n , v n ) ( u n , v n ) F ( u n , v n ) ] [ 1 2 F ( u ˇ , v ˇ ) ( u ˇ , v ˇ ) F ( u ˇ , v ˇ ) ] + o n ( 1 ) = Φ ( u ˇ , v ˇ ) + o n ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ14_HTML.gif
(4.3)

where (4.3) follows from the Fatou lemma and (2.3). Then Φ ( u ˇ , v ˇ ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq199_HTML.gif. According to ( u ˇ , v ˇ ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq198_HTML.gif, we have Φ ( u ˇ , v ˇ ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq200_HTML.gif. Thus, Φ ( u ˇ , v ˇ ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq201_HTML.gif. Consequently, ( u ˇ , v ˇ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq202_HTML.gif is a ground state of (NLS).

It remains to look for a positive ground state for (NLS). First, we can assume that ( u ˇ , v ˇ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq202_HTML.gif is non-negative. In fact, note that | u | 2 2 = | | u | | 2 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq203_HTML.gif and | v | 2 2 = | | v | | 2 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq204_HTML.gif for all ( u , v ) H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq205_HTML.gif. Then ( | u ˇ | , | v ˇ | ) H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq206_HTML.gif. Let τ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq207_HTML.gif be such that τ ( | u ˇ | , | v ˇ | ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq208_HTML.gif. By (F6) we easily have that Φ ( τ | u ˇ | , τ | v ˇ | ) Φ ( τ u ˇ , τ v ˇ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq209_HTML.gif. Moreover, Φ ( τ u ˇ , τ v ˇ ) Φ ( u ˇ , v ˇ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq210_HTML.gif since ( u ˇ , v ˇ ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq211_HTML.gif. Then Φ ( τ | u ˇ | , τ | v ˇ | ) Φ ( u ˇ , v ˇ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq212_HTML.gif. So, ( τ | u ˇ | , τ | v ˇ | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq213_HTML.gif is also a minimizer of Φ on M. Then ( τ | u ˇ | , τ | v ˇ | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq213_HTML.gif is also a ground state of (NLS). Thus we can assume that ( u ˇ , v ˇ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq202_HTML.gif is a non-negative ground state for (NLS). Now, we claim that u ˇ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq197_HTML.gif, v ˇ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq214_HTML.gif. Indeed, if u ˇ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq215_HTML.gif, then from (F5) and λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq18_HTML.gif, the first equation of (NLS) yields that v ˇ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq216_HTML.gif. Then ( u ˇ , v ˇ ) = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq217_HTML.gif. This is impossible. So, u ˇ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq197_HTML.gif. Similarly, v ˇ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq214_HTML.gif. By (F5), applying the maximum principle to each equation of (NLS), we infer that u ˇ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq218_HTML.gif, v ˇ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq219_HTML.gif. The proof is complete. □

5 The asymptotically periodic case

In this section, we will consider the asymptotically periodic case and prove Theorems 1.2 and 1.3. As in the proof of Theorem 1.1, we first take advantage of the minimizing sequence of Ψ to find a ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq57_HTML.gif sequence of Φ. In what follows, the important thing is to recover the compactness for the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq57_HTML.gif sequence. For this purpose, we need to estimate the functional levels of the problem (NLS) and those of a related periodic problem of (NLS) (roughly speaking, the limit system of (NLS) by (V3))
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equy_HTML.gif
Hence, first we introduce some definitions and look for solutions for the problem (NLS) p . The functional of (NLS) p is defined by
Φ p ( u , v ) = 1 2 ( u 2 + v 2 + a p ( x ) u 2 + b p ( x ) v 2 ) λ u v F ( u , v ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equz_HTML.gif
The Nehari manifold of (NLS) p is
M p = { ( u , v ) H { ( 0 , 0 ) } : Φ p ( u , v ) , ( u , v ) = 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equaa_HTML.gif
and c p = inf M p Φ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq220_HTML.gif is the least energy of (NLS) p on M p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq221_HTML.gif. Note that
Φ p | M p ( u , v ) = [ 1 2 F ( u , v ) ( u , v ) F ( u , v ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ15_HTML.gif
(5.1)

As for c, we have c p 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq222_HTML.gif.

Lemma 5.1 Suppose that a p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq50_HTML.gif, b p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq51_HTML.gif satisfy (V1) and (V2). Let (F1)-(F6) hold. Then the problem (NLS) p has a positive ground state ( u , v ) M p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq223_HTML.gif such that Φ p ( u , v ) = c p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq224_HTML.gif.

Proof As a corollary of Theorem 1.1, we infer that the problem (NLS) p has a positive ground state. Moreover, from the argument of Theorem 1.1, we find that the ground state of the problem (NLS) p we obtained is a minimizer of Φ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq225_HTML.gif on M p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq221_HTML.gif. □

The existence of a positive ground state for the problem (NLS) p implies that (NLS) has a positive ground state when a = a p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq226_HTML.gif and b = b p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq227_HTML.gif. So, it remains to consider
a a p or b b p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ16_HTML.gif
(5.2)

Next, we prove that c < c p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq228_HTML.gif under some conditions.

Lemma 5.2 Suppose that a p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq50_HTML.gif, b p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq51_HTML.gif satisfy (V2). Let (V1), (V4), (5.2) and (F1)-(F6) hold. Then c < c p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq228_HTML.gif.

Proof Let ( u 0 , v 0 ) M p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq229_HTML.gif be a positive ground state of (NLS) p such that Φ p ( u 0 , v 0 ) = c p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq230_HTML.gif. Assume t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq231_HTML.gif satisfies t ( u 0 , v 0 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq232_HTML.gif. By (V4), we get
[ ( a ( x ) a p ( x ) ) u 0 2 + ( b ( x ) b p ( x ) ) v 0 2 ] 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equab_HTML.gif

Then Φ ( t u 0 , t v 0 ) Φ p ( t u 0 , t v 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq233_HTML.gif.

Replacing Φ and M by Φ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq225_HTML.gif and M p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq221_HTML.gif respectively, (3.3) also holds. Noting that ( u 0 , v 0 ) M p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq229_HTML.gif, we infer that
Φ p ( t u 0 , t v 0 ) Φ p ( u 0 , v 0 ) and Φ p ( t u 0 , t v 0 ) = Φ p ( u 0 , v 0 ) if and only if  t = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ17_HTML.gif
(5.3)
Therefore,
c Φ ( t u 0 , t v 0 ) Φ p ( t u 0 , t v 0 ) Φ p ( u 0 , v 0 ) = c p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ18_HTML.gif
(5.4)

If c < c p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq228_HTML.gif, we are done. Otherwise, c = c p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq234_HTML.gif. Then by (5.3) and (5.4), we get t = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq235_HTML.gif and Φ ( u 0 , v 0 ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq236_HTML.gif. Then ( u 0 , v 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq237_HTML.gif is a ground state for (NLS). Note that ( u 0 , v 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq237_HTML.gif is a solution of (NLS) p . From the first equations of (NLS) and (NLS) p , we infer that a = a p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq226_HTML.gif. Similarly, b = b p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq227_HTML.gif contrary to (5.2). The proof is now complete. □

Lemma 5.3 Suppose that a p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq50_HTML.gif, b p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq51_HTML.gif satisfy (V1) and (V2). Let (V1), (V5), (5.2) and (F1)-(F7) hold. Then c < c p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq228_HTML.gif.

Proof Let ( u 0 , v 0 ) M p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq229_HTML.gif be a positive ground state of (NLS) p such that Φ p ( u 0 , v 0 ) = c p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq230_HTML.gif. By (V5) and (F7), we find that ( v 0 , u 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq238_HTML.gif is also a minimizer of Φ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq225_HTML.gif on M p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq221_HTML.gif. Let t , τ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq239_HTML.gif be such that t ( u 0 , v 0 ) , τ ( v 0 , u 0 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq240_HTML.gif. Using (V5), we have ( a ( x ) + b ( x ) 2 V ( x ) ) ( u 0 2 + v 0 2 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq241_HTML.gif. Then
[ ( a ( x ) V ( x ) ) u 0 2 + ( b ( x ) V ( x ) ) v 0 2 ] 0 or [ ( a ( x ) V ( x ) ) v 0 2 + ( b ( x ) V ( x ) ) u 0 2 ] 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equac_HTML.gif
Without loss of generality, we assume that
[ ( a ( x ) V ( x ) ) u 0 2 + ( b ( x ) V ( x ) ) v 0 2 ] 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equad_HTML.gif

Then Φ ( t u 0 , t v 0 ) Φ p ( t u 0 , t v 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq233_HTML.gif. Below we argue analogously with the proof of Lemma 5.2 to infer that c < c p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq228_HTML.gif. This ends the proof. □

Now, we are ready to prove Theorems 1.2 and 1.3. The proof is partially inspired by [17], where the authors dealt with Schrödinger-Poisson equations.

Proof of Theorem 1.2 As the argument of Theorem 1.1, we infer that there exists a sequence ( u n , v n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq189_HTML.gif such that Φ ( u n , v n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq153_HTML.gif and Φ ( u n , v n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq242_HTML.gif.

We claim that { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq108_HTML.gif is bounded in H. Otherwise, suppose t n : = ( u n , v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq243_HTML.gif up to a subsequence. Set ( w n , z n ) = ( u n , v n ) ( u n , v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq157_HTML.gif. As in the proof of Theorem 1.1, taking a subsequence, we suppose | w n | q + | z n | q A [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq244_HTML.gif and exclude the case that A = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq163_HTML.gif. So, A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq169_HTML.gif, then we can assume that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq245_HTML.gif in L q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq171_HTML.gif. From the Lions compactness lemma, it follows that there exist δ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq172_HTML.gif and y n R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq246_HTML.gif such that
B 1 ( y n ) | w n | 2 > δ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equae_HTML.gif
Set w ˜ n = w n ( + y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq247_HTML.gif and z ˜ n = z n ( + y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq248_HTML.gif. We assume that ( w ˜ n , z ˜ n ) ( w ˜ , z ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq249_HTML.gif in H, ( w ˜ n , z ˜ n ) ( w ˜ , z ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq250_HTML.gif in L loc 2 ( R N ) × L loc 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq102_HTML.gif and ( w ˜ n , z ˜ n ) ( w ˜ , z ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq250_HTML.gif a.e. on R 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq103_HTML.gif up to a subsequence. Then by
B 1 ( y n ) | w n | 2 > δ 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equaf_HTML.gif
we obtain w ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq251_HTML.gif. So, Lemma 3.2 implies that
F ( t n w ˜ n , t n z ˜ n ) t n 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equag_HTML.gif
Then by (2.1), we get
0 Φ ( u n , v n ) ( u n , v n ) 2 = 1 2 [ u n 2 + v n 2 + a ( x ) u n 2 + b ( x ) v n 2 2 λ u n v n ( u n , v n ) 2 ] F ( t n w n , t n z n ) t n 2 ν 2 F ( t n w ˜ n , t n z ˜ n ) t n 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equah_HTML.gif

This is a contradiction.

Hence, { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq108_HTML.gif is bounded in H. Up to a subsequence, we assume that ( u n , v n ) ( u ˜ , v ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq252_HTML.gif in H, ( u n , v n ) ( u ˜ , v ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq253_HTML.gif in L loc 2 ( R N ) × L loc 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq102_HTML.gif and ( u n , v n ) ( u ˜ , v ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq253_HTML.gif a.e. on R 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq103_HTML.gif. By Lemma 2.2, we have Φ ( u ˜ , v ˜ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq254_HTML.gif. Namely, ( u ˜ , v ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq76_HTML.gif is a solution of (NLS).

Below we prove that ( u ˜ , v ˜ ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq255_HTML.gif. We argue by contradiction. Suppose that ( u ˜ , v ˜ ) = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq256_HTML.gif. By (V3), for any ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq77_HTML.gif, there exists r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq257_HTML.gif such that
| a ( x ) a p ( x ) | < ϵ , | b ( x ) b p ( x ) | < ϵ for all  | x | > r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ19_HTML.gif
(5.5)
Note that ( u ˜ , v ˜ ) = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq256_HTML.gif, after passing to a subsequence, we assume ( u n , v n ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq188_HTML.gif in L 2 ( B r ( 0 ) ) × L 2 ( B r ( 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq258_HTML.gif. So, for the above ϵ, there exists J 1 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq259_HTML.gif such that for n > J 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq260_HTML.gif, we have
B r ( 0 ) u n 2 < ϵ , B r ( 0 ) v n 2 < ϵ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equai_HTML.gif
Combining with (5.5), for n > J 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq260_HTML.gif, we get
| ( a ( x ) a p ( x ) ) u n 2 | B r ( 0 ) | a ( x ) a p ( x ) | u n 2 + ϵ R N B r ( 0 ) u n 2 < ( | a | + | a p | ) ϵ + C ϵ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equaj_HTML.gif
Then ( a ( x ) a p ( x ) ) u n 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq261_HTML.gif. Similarly, ( b ( x ) b p ( x ) ) v n 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq262_HTML.gif. Therefore,
Φ p ( u n , v n ) = Φ ( u n , v n ) + o n ( 1 ) , Φ p ( u n , v n ) , ( u n , v n ) = Φ ( u n , v n ) , ( u n , v n ) + o n ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equak_HTML.gif
Hence,
Φ p ( u n , v n ) = c + o n ( 1 ) , Φ p ( u n , v n ) , ( u n , v n ) = o n ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ20_HTML.gif
(5.6)

Let s n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq263_HTML.gif be such that s n ( u n , v n ) M p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq264_HTML.gif. We claim that s n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq265_HTML.gif for large n and s n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq266_HTML.gif.

First, we prove that
lim sup n s n 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ21_HTML.gif
(5.7)
Otherwise, there exist δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq267_HTML.gif and a subsequence of s n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq268_HTML.gif, still denoted by s n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq268_HTML.gif, such that s n 1 + δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq269_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq270_HTML.gif. From (5.6) we have
u n 2 + v n 2 + a p ( x ) u n 2 + b p ( x ) v n 2 2 λ u n v n = F ( u n , v n ) ( u n , v n ) + o n ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equal_HTML.gif
Moreover, by s n ( u n , v n ) M p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq264_HTML.gif, we get
s n 2 [ u n 2 + v n 2 + a p ( x ) u n 2 + b p ( x ) v n 2 2 λ u n v n ] = F ( s n u n , s n v n ) ( s n u n , s n v n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equam_HTML.gif
Hence,
[ F ( s n u n , s n v n ) ( u n , v n ) s n F ( u n , v n ) ( u n , v n ) ] = o n ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equan_HTML.gif
By s n 1 + δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq269_HTML.gif and (F3), we obtain
[ F ( ( 1 + δ ) u n , ( 1 + δ ) v n ) ( u n , v n ) 1 + δ F ( u n , v n ) ( u n , v n ) ] o n ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ22_HTML.gif
(5.8)
Similar to the proof of Theorem 1.1, if ( u n , v n ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq188_HTML.gif in L q ( R N ) × L q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq160_HTML.gif, then ( u n , v n ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq188_HTML.gif in H. Contrary to (3.2), since ( u n , v n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq189_HTML.gif, therefore, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq271_HTML.gif in L q ( R N ) × L q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq160_HTML.gif. Suppose https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq272_HTML.gif in L q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq171_HTML.gif. Then from the Lions compactness lemma, it follows that there exist x n R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq273_HTML.gif and δ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq274_HTML.gif such that
B 1 ( x n ) u n 2 > δ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ23_HTML.gif
(5.9)
We denote u ¯ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq275_HTML.gif and v ¯ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq276_HTML.gif by u ¯ n = u n ( + x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq277_HTML.gif and v ¯ n = v n ( + x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq278_HTML.gif. Similarly, we assume that ( u ¯ n , v ¯ n ) ( u ¯ , v ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq279_HTML.gif in H, ( u ¯ n , v ¯ n ) ( u ¯ , v ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq280_HTML.gif in L loc 2 ( R N ) × L loc 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq102_HTML.gif and ( u ¯ n , u ¯ n ) ( u ¯ , v ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq281_HTML.gif a.e. on R 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq103_HTML.gif up to a subsequence. By (5.9), we have
B 1 ( 0 ) u ¯ n 2 > δ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equao_HTML.gif
So, u ¯ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq282_HTML.gif. From (5.8), (F3) and the Fatou lemma, we obtain
0 < [ F ( ( 1 + δ ) u ¯ , ( 1 + δ ) v ¯ ) ( u ¯ , v ¯ ) 1 + δ F ( u ¯ , v ¯ ) ( u ¯ , v ¯ ) ] 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equap_HTML.gif

which is impossible. Consequently, (5.7) holds.

Now, we show that s n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq265_HTML.gif for large n. Indeed, on the contrary, passing to a subsequence, we assume that s n < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq283_HTML.gif. Using (3.1) and (5.1), we have
c p Φ p ( s n u n , s n v n ) = [ 1 2 F ( s n u n , s n v n ) ( s n u n , s n v n ) F ( s n u n , s n v n ) ] [ 1 2 F ( u n , v n ) ( u n , v n ) F ( u n , v n ) ] = Φ ( u n , v n ) + o n ( 1 ) = c + o n ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ24_HTML.gif
(5.10)
where (5.10) follows from the fact that α is increasing in ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq284_HTML.gif by Lemma 2.1. Then c p c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq285_HTML.gif, contrary to Lemma 5.2. Therefore, combining with (5.7), we may assume that
s n 1 , for large  n and lim n s n = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ25_HTML.gif
(5.11)
For ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq77_HTML.gif and 1 s s n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq286_HTML.gif, using (2.2) we get
| F ( s u n , s v n ) ( u n , v n ) | ϵ s n ( | u n | 2 2 + | v n | 2 2 ) + C ϵ s n q 1 ( | u n | q q + | v n | q q ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ26_HTML.gif
(5.12)
Combining (5.11) with (5.12), one easily has that
[ F ( s n u n , s n v n ) F ( u n , v n ) ] = 1 s n [ F ( s u n , s v n ) ( u n , v n ) d x ] d s = o n ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equaq_HTML.gif
Since a p , b p L ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq45_HTML.gif and { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq108_HTML.gif is bounded, we get
s n 2 1 2 [ u n 2 + v n 2 + a p ( x ) u n 2 + b p ( x ) v n 2 2 λ u n v n ] = o n ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equar_HTML.gif

Hence, Φ p ( s n u n , s n v n ) = Φ p ( u n , v n ) + o n ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq287_HTML.gif. Then using (5.6), we have c p Φ p ( s n u n , s n v n ) = c + o n ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq288_HTML.gif. Then c p c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq285_HTML.gif. However, Lemma 5.2 implies that c < c p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq228_HTML.gif. This is a contradiction. Note that this contradiction follows from the hypothesis that ( u ˜ , v ˜ ) = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq256_HTML.gif. So, ( u ˜ , v ˜ ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq255_HTML.gif. Then ( u ˜ , v ˜ ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq289_HTML.gif.

It suffices to show that Φ ( u ˜ , v ˜ ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq290_HTML.gif. By (3.1) we have
c + o n ( 1 ) = Φ ( u n , v n ) = [ 1 2 F ( u n , v n ) ( u n , v n ) F ( u n , v n ) ] [ 1 2 F ( u ˜ , v ˜ ) ( u ˜ , v ˜ ) F ( u ˜ , v ˜ ) ] + o n ( 1 ) = Φ ( u ˜ , v ˜ ) + o n ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_Equ27_HTML.gif
(5.13)

where the inequality (5.13) holds by (2.3) and the Fatou lemma. Then Φ ( u ˜ , v ˜ ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq291_HTML.gif. According to ( u ˜ , v ˜ ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq289_HTML.gif, we have Φ ( u ˜ , v ˜ ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq290_HTML.gif. Then ( u ˜ , v ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-13/MediaObjects/13661_2012_Article_270_IEq76_HTML.gif is a ground state for (NLS). Below we argue analogously with the proof of Theorem 1.1 to get a positive ground state for (NLS). The proof is complete. □

Proof of Theorem 1.3 By Lemma 5.3, repeating the argument of Theorem 1.2, we show the existence of a ground state for (NLS) and then look for a positive ground state as the argument of Theorem 1.1. □

Declarations

Acknowledgements

The authors would like to express their sincere gratitude to the referee for helpful and insightful comments. Hui Zhang was supported by the Research and Innovation Project for College Graduates of Jiangsu Province with contract number CXLX12_0069, Junxiang Xu and Fubao Zhang were supported by the National Natural Science Foundation of China with contract number 11071038.

Authors’ Affiliations

(1)
Department of Mathematics, Southeast University

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