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Existence of positive ground states for some nonlinear Schrödinger systems

Hui Zhang, Junxiang Xu* and Fubao Zhang

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Department of Mathematics, Southeast University, Nanjing, 210096, P.R. China

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Citation and License

Boundary Value Problems 2013, 2013:13  doi:10.1186/1687-2770-2013-13


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/13


Received:26 March 2012
Accepted:12 January 2013
Published:28 January 2013

© 2013 Zhang et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M1">View MathML</a>

where a, b are periodic or asymptotically periodic and F satisfies some superlinear conditions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M2">View MathML</a>. 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M3">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M7">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M9">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M10">View MathML</a>, the system (1.1) turns out to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M11">View MathML</a>

(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 (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M15">View MathML</a> and β); see [3-9] and the references therein. Later in [10], assuming <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M16">View MathML</a>, Pomponio and Secchi established the existence of radially symmetric ground states for (1.2) with general nonlinearities (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M18">View MathML</a>).

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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M19">View MathML</a>

(1.3)

where G is periodic in x and satisfies some superlinear conditions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M2">View MathML</a>. For non-periodic case, we refer to [8,12-14] 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M21">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M22">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M23">View MathML</a>

(NLS)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M24">View MathML</a> is a real parameter. For simplicity, we denote +∞ by ∞

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M25">View MathML</a>

Moreover, in what follows, the notation inf (sup) is understood as the essential infimum (supremum). In the sequel, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M27">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M28">View MathML</a>, we always assume that

(V1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M30">View MathML</a>,

(F1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M31">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M33">View MathML</a>,

(F2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M34">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M35">View MathML</a>,

(F3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M38">View MathML</a> is strictly increasing,

(F4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M39">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M40">View MathML</a>,

(F5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M45">View MathML</a>,

(F6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M47">View MathML</a>.

(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 onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M51">View MathML</a>.

We have the following result.

Theorem 1.1Let (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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M52">View MathML</a>

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 onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M53">View MathML</a> satisfying (V1) and (V2) and a, b satisfies that

(V3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M55">View MathML</a>,

(V4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M57">View MathML</a>.

We have the following result.

Theorem 1.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59">View MathML</a>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 onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59">View MathML</a> satisfy (V1).

In addition, we consider the following conditions:

(V5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M63">View MathML</a>,

(F7) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M66">View MathML</a>.

We have the following result.

Theorem 1.3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59">View MathML</a>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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M21">View MathML</a> to find a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70">View MathML</a> sequence. When the system (NLS) is asymptotically periodic, the difficulty is to recover the compactness for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M74">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M75">View MathML</a> is measurable.

X denotes the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M76">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M22">View MathML</a>), with the standard scalar product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M78">View MathML</a> and the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M79">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M80">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M81">View MathML</a>. When there is no possible misunderstanding, the subscripts could be omitted.

• The usual norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M82">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M83">View MathML</a>) will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M84">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M85">View MathML</a>.

• For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M88">View MathML</a> denotes the ball of radius ϱ centered at z.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M27">View MathML</a>. Then by conditions (F1) and (F2), the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M91">View MathML</a>

is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M92">View MathML</a> and its critical points are solutions of (NLS). Moreover, by (V1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M93">View MathML</a>

(2.1)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M94">View MathML</a>.

A solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M95">View MathML</a> of (NLS) is called a ground state if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M96">View MathML</a>

A ground state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M2">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M66">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M45">View MathML</a>) 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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M102">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M103">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M104">View MathML</a>

(2.2)

(F2) and (F3) yield that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M105">View MathML</a>

(2.3)

Moreover, (F3) implies the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M106">View MathML</a>is increasing in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M107">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M108">View MathML</a>.

Proof The inequalities (2.2) and (2.3) are easily inferred from the corresponding assumptions. We just prove the last conclusion. Indeed, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M109">View MathML</a>. Then by (F3) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M110">View MathML</a>

 □

Lemma 2.2Let (F1) and (F2) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M111">View MathML</a>is weakly sequentially continuous. Namely, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M112">View MathML</a>inH, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M113">View MathML</a>inH.

Proof Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M112">View MathML</a> in H. After passing to a subsequence, we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M115">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M116">View MathML</a>. By (F1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M117">View MathML</a>

Then for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M118">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M119">View MathML</a>

So, one easily has that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M120">View MathML</a>

(2.4)

Now, we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M121">View MathML</a> is bounded in H. Indeed, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M122">View MathML</a>, using (2.2) and the Hölder inequality, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M123">View MathML</a>

Similarly, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M124">View MathML</a>. Then we easily have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M125">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M121">View MathML</a> is bounded in H. Combining with the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M127">View MathML</a> is dense in H, we easily deduce that (2.4) holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M128">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M113">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M130">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M131">View MathML</a>

Below we investigate the main properties of Φ on M.

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

Proof

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M132">View MathML</a>

(3.1)

By (2.3) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M133">View MathML</a>. □

Define the least energy of (NLS) on M by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M134">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M135">View MathML</a>. 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 3.2Let (V1) and (F2)-(F4) hold. Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M112">View MathML</a>in H and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M137">View MathML</a>. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M138">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M139">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M140">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M141">View MathML</a>

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M142">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M112">View MathML</a> in H, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M115">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M115">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147">View MathML</a> for a subsequence. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M137">View MathML</a>, there exists a positive measure set Ω such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M150">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M152">View MathML</a>. By (F4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M153">View MathML</a>

Therefore, (2.3) and the Fatou lemma yield that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M154">View MathML</a>

Using (2.1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M155">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156">View MathML</a> is bounded in H. □

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

(i) for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M157">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M158">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M159">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M160">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M161">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M162">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M163">View MathML</a>;

(ii) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M164">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M165">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M166">View MathML</a>;

(iii) for each compact subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M167">View MathML</a>, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M168">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M169">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M170">View MathML</a>.

Proof

(i) Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M171">View MathML</a>

Using (F2), we infer that when t is small enough, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M160">View MathML</a>. By Lemma 3.2 and (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M173">View MathML</a>

Then when t is large enough, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M162">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M175">View MathML</a> has maximum points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M107">View MathML</a>. Moreover, from (F3) one easily deduces that the critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M175">View MathML</a> is unique in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M107">View MathML</a>, and then it is the maximum point. We denote it by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M158">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M160">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M162">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M163">View MathML</a>.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M184">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M185">View MathML</a>

By (2.1) and (2.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M186">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M102">View MathML</a> is arbitrary. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M188">View MathML</a>

So, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M164">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M190">View MathML</a>

(3.2)

Using (i), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M166">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M192">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M193">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M194">View MathML</a>. Then (3.2) yields the conclusion (ii).

(iii) We argue by contradiction. Suppose that there exist a compact set W and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M197">View MathML</a>. Since W is compact, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M170">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M115">View MathML</a> in H. Then Lemma 3.2 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M200">View MathML</a>. Contrary to Lemma 3.1 since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M201">View MathML</a>. This ends the proof. □

Remark 3.1 Lemma 3.3(i) implies that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M157">View MathML</a>, there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M203">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M204">View MathML</a>

(3.3)

As a consequence of Lemma 3.3(i), we can define the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M205">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M206">View MathML</a>. By Lemma 3.3, [[11], Proposition 3.1(b)] yields the following result.

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

If M is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M92">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M208">View MathML</a> and solve the minimizing problem. However, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M27">View MathML</a>, M may not be a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M92">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M211">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M212">View MathML</a>, and we have the following conclusion.

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

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M213">View MathML</a>is aPSsequence for Ψ, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M214">View MathML</a>is aPSsequence for Φ.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M215">View MathML</a>is a critical point of Ψ if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M216">View MathML</a>is a nontrivial critical point of Φ. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M217">View MathML</a>.

(iii) A minimizer of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M208">View MathML</a>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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M184">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M220">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M221">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M222">View MathML</a>. By the conclusion (ii), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M223">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M224">View MathML</a>. Using the conclusion (ii) again, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M225">View MathML</a>. □

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M208">View MathML</a>. By Proposition 3.1(iii), in order to look for a ground state for (NLS), we just need to seek for a minimizer of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M208">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70">View MathML</a> sequence of Φ. Then we use the invariant of the functional under translation of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M230">View MathML</a> to recover the compactness for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70">View MathML</a> sequence.

Proof of Theorem 1.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M232">View MathML</a> be a minimizing sequence of Ψ. By the Ekeland variational principle [[16], Theorem 8.5], we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M233">View MathML</a>. Using Proposition 3.1(i), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M234">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M235">View MathML</a>. Proposition 3.1(ii) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M236">View MathML</a>.

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156">View MathML</a> is bounded in H. Otherwise, suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M238">View MathML</a> up to a subsequence. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M239">View MathML</a>. Then we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M240">View MathML</a> in H, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M241">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M241">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147">View MathML</a> after passing to a subsequence. Moreover, the Sobolev embedding theorem implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M213">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M246">View MathML</a>, namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M247">View MathML</a> is bounded. Taking a subsequence, we suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M248">View MathML</a>.

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M249">View MathML</a>, then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M102">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M251">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M252">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M253">View MathML</a>. Combining with (2.2), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M253">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M37">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M256">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M257">View MathML</a>. Hence, by (2.1) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M258">View MathML</a>

a contradiction for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M259">View MathML</a>.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M260">View MathML</a>, then we can assume that in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M262">View MathML</a>. From the Lions compactness lemma [[16], Lemma 1.21], it follows that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M263">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M264">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M265">View MathML</a>

(4.1)

Since Φ and M are invariant by translation of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M51">View MathML</a>, translating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M268">View MathML</a> if necessary, we may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M269">View MathML</a> is bounded. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M270">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M271">View MathML</a>, then (4.1) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M272">View MathML</a>. Then from Lemma 3.2, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M273">View MathML</a>. This is impossible since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M274">View MathML</a>.

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156">View MathML</a> is bounded in H. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M276">View MathML</a> in H, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M277">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M277">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147">View MathML</a> for a subsequence. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M234">View MathML</a>, Lemma 2.2 yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M282">View MathML</a>.

We will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M283">View MathML</a>. Similarly, suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M284">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M285">View MathML</a>, then as before, combining with (2.2), we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M286">View MathML</a>. Hence, by (2.1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M287">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M288">View MathML</a> in H. This is impossible since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M289">View MathML</a> and (3.2) holds. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M290">View MathML</a>. So, we can assume in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M262">View MathML</a>. Then the Lions compactness lemma implies that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M294">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M295">View MathML</a>

(4.2)

As before, translating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M296">View MathML</a> if necessary, we may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M297">View MathML</a> is bounded. Since (4.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M298">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M271">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M300">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M282">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M302">View MathML</a>. Then by (3.1) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M303">View MathML</a>

(4.3)

where (4.3) follows from the Fatou lemma and (2.3). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M304">View MathML</a>. According to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M302">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M306">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M307">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M308">View MathML</a> is a ground state of (NLS).

It remains to look for a positive ground state for (NLS). First, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M308">View MathML</a> is non-negative. In fact, note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M310">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M311">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M312">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M313">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M314">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M315">View MathML</a>. By (F6) we easily have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M316">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M317">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M318">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M319">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M320">View MathML</a> is also a minimizer of Φ on M. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M320">View MathML</a> is also a ground state of (NLS). Thus we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M308">View MathML</a> is a non-negative ground state for (NLS). Now, we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M300">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M324">View MathML</a>. Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M325">View MathML</a>, then from (F5) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M24">View MathML</a>, the first equation of (NLS) yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M327">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M328">View MathML</a>. This is impossible. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M300">View MathML</a>. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M324">View MathML</a>. By (F5), applying the maximum principle to each equation of (NLS), we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M331">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M332">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70">View MathML</a> sequence of Φ. In what follows, the important thing is to recover the compactness for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M70">View MathML</a> 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))

Hence, first we introduce some definitions and look for solutions for the problem (NLS)p. The functional of (NLS)p is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M336">View MathML</a>

The Nehari manifold of (NLS)p is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M337">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M338">View MathML</a> is the least energy of (NLS)p on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M339">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M340">View MathML</a>

(5.1)

As for c, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M341">View MathML</a>.

Lemma 5.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59">View MathML</a>satisfy (V1) and (V2). Let (F1)-(F6) hold. Then the problem (NLS)phas a positive ground state<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M344">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M345">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M346">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M339">View MathML</a>. □

The existence of a positive ground state for the problem (NLS)p implies that (NLS) has a positive ground state when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M348">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M349">View MathML</a>. So, it remains to consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M350">View MathML</a>

(5.2)

Next, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351">View MathML</a> under some conditions.

Lemma 5.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59">View MathML</a>satisfy (V2). Let (V1), (V4), (5.2) and (F1)-(F6) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M355">View MathML</a> be a positive ground state of (NLS)p such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M356">View MathML</a>. Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M357">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M358">View MathML</a>. By (V4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M359">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M360">View MathML</a>.

Replacing Φ and M by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M346">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M339">View MathML</a> respectively, (3.3) also holds. Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M355">View MathML</a>, we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M364">View MathML</a>

(5.3)

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M365">View MathML</a>

(5.4)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351">View MathML</a>, we are done. Otherwise, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M367">View MathML</a>. Then by (5.3) and (5.4), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M368">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M369">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M370">View MathML</a> is a ground state for (NLS). Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M370">View MathML</a> is a solution of (NLS)p. From the first equations of (NLS) and (NLS)p, we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M348">View MathML</a>. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M349">View MathML</a> contrary to (5.2). The proof is now complete. □

Lemma 5.3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M59">View MathML</a>satisfy (V1) and (V2). Let (V1), (V5), (5.2) and (F1)-(F7) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M355">View MathML</a> be a positive ground state of (NLS)p such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M356">View MathML</a>. By (V5) and (F7), we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M379">View MathML</a> is also a minimizer of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M346">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M339">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M382">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M383">View MathML</a>. Using (V5), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M384">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M385">View MathML</a>

Without loss of generality, we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M386">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M360">View MathML</a>. Below we argue analogously with the proof of Lemma 5.2 to infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M289">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M391">View MathML</a>.

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156">View MathML</a> is bounded in H. Otherwise, suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M393">View MathML</a> up to a subsequence. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M239">View MathML</a>. As in the proof of Theorem 1.1, taking a subsequence, we suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M395">View MathML</a> and exclude the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M249">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M260">View MathML</a>, then we can assume that in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M262">View MathML</a>. From the Lions compactness lemma, it follows that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M263">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M401">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M402">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M403">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M404">View MathML</a>. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M405">View MathML</a> in H, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M406">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M406">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147">View MathML</a> up to a subsequence. Then by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M410">View MathML</a>

we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M411">View MathML</a>. So, Lemma 3.2 implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M412">View MathML</a>

Then by (2.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M413">View MathML</a>

This is a contradiction.

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156">View MathML</a> is bounded in H. Up to a subsequence, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M415">View MathML</a> in H, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M416">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M416">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147">View MathML</a>. By Lemma 2.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M420">View MathML</a>. Namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M95">View MathML</a> is a solution of (NLS).

Below we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M422">View MathML</a>. We argue by contradiction. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M423">View MathML</a>. By (V3), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M102">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M425">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M426">View MathML</a>

(5.5)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M423">View MathML</a>, after passing to a subsequence, we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M288">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M429">View MathML</a>. So, for the above ϵ, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M430">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M431">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M432">View MathML</a>

Combining with (5.5), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M431">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M434">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M435">View MathML</a>. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M436">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M437">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M438">View MathML</a>

(5.6)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M439">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M440">View MathML</a>. We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M441">View MathML</a> for large n and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M442">View MathML</a>.

First, we prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M443">View MathML</a>

(5.7)

Otherwise, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M444">View MathML</a> and a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M445">View MathML</a>, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M445">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M447">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M448">View MathML</a>. From (5.6) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M449">View MathML</a>

Moreover, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M440">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M451">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M452">View MathML</a>

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M447">View MathML</a> and (F3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M454">View MathML</a>

(5.8)

Similar to the proof of Theorem 1.1, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M288">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M246">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M288">View MathML</a> in H. Contrary to (3.2), since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M289">View MathML</a>, therefore, in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M246">View MathML</a>. Suppose in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M262">View MathML</a>. Then from the Lions compactness lemma, it follows that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M463">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M464">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M465">View MathML</a>

(5.9)

We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M466">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M467">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M468">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M469">View MathML</a>. Similarly, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M470">View MathML</a> in H, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M471">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M145">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M473">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M147">View MathML</a> up to a subsequence. By (5.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M475">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M476">View MathML</a>. From (5.8), (F3) and the Fatou lemma, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M477">View MathML</a>

which is impossible. Consequently, (5.7) holds.

Now, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M441">View MathML</a> for large n. Indeed, on the contrary, passing to a subsequence, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M479">View MathML</a>. Using (3.1) and (5.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M480">View MathML</a>

(5.10)

where (5.10) follows from the fact that α is increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M481">View MathML</a> by Lemma 2.1. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M482">View MathML</a>, contrary to Lemma 5.2. Therefore, combining with (5.7), we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M483">View MathML</a>

(5.11)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M485">View MathML</a>, using (2.2) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M486">View MathML</a>

(5.12)

Combining (5.11) with (5.12), one easily has that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M487">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M156">View MathML</a> is bounded, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M490">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M491">View MathML</a>. Then using (5.6), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M492">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M482">View MathML</a>. However, Lemma 5.2 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M351">View MathML</a>. This is a contradiction. Note that this contradiction follows from the hypothesis that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M423">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M422">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M497">View MathML</a>.

It suffices to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M498">View MathML</a>. By (3.1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M499">View MathML</a>

(5.13)

where the inequality (5.13) holds by (2.3) and the Fatou lemma. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M500">View MathML</a>. According to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M497">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M498">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/13/mathml/M95">View MathML</a> 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. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The paper is a joint work of all the authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.

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.

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