Open Access Research

Infinitely many periodic solutions for some second-order differential systems with p(t)-Laplacian

Liang Zhang, Xian Hua Tang* and Jing Chen

Author Affiliations

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, P. R. China

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Boundary Value Problems 2011, 2011:33  doi:10.1186/1687-2770-2011-33


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/33


Received:3 June 2011
Accepted:14 October 2011
Published:14 October 2011

© 2011 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

In this article, we investigate the existence of infinitely many periodic solutions for some nonautonomous second-order differential systems with p(t)-Laplacian. Some multiplicity results are obtained using critical point theory.

2000 Mathematics Subject Classification: 34C37; 58E05; 70H05.

Keywords:
p(t)-Laplacian; Periodic solutions; Critical point theory

1. Introduction

Consider the second-order differential system with p(t)-Laplacian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M1">View MathML</a>

(1.1)

where T > 0, F: [0, T] × ℝN → ℝ, and p(t) ∈ C([0, T], ℝ+) satisfies the following assumptions:

(A) p(0) = p(T) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M2">View MathML</a>, where q+ > 1 which satisfies 1/p- + 1/q+ = 1.

Moreover, we suppose that F: [0, T] × ℝN → ℝ satisfies the following assumptions:

(A') F(t, x) is measurable in t for every x ∈ ℝN and continuously differentiable in x for a.e. t ∈ [0, T], and there exist a C(ℝ+, ℝ+), b L1(0, T; ℝ+), such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M3">View MathML</a>

for all x ∈ ℝN and a.e. t ∈ [0, T].

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M4">View MathML</a> is said to be p(t)-Laplacian, and becomes p-Laplacian when p(t) ≡ p (a constant). The p(t)-Laplacian possesses more complicated nonlinearity than p-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth conditions has received considerable attention in recent years. These problems are interesting in applications and raise many mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field. Another field of application of equations with variable exponent growth conditions is image processing (see [1,2]). The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [3-12] for an overview on this subject.

In 2003, Fan and Fan [13] studied the ordinary p(t)-Laplacian system and introduced a generalized Orlicz-Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a>, which is different from the usual space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M6">View MathML</a>, then Wang and Yuan [14] obtained the existence and multiplicity of periodic solutions for ordinary p(t)-Laplacian system under the generalized Ambrosetti-Rabinowitz conditions. Fountain and Dual Fountain theorems were established by Bartsch and Willem [15,16], and both theorems are effective tools for studying the existence of infinitely many large energy solutions and small energy solutions. When we impose some suitable conditions on the growth of the potential function at origin or at infinity, we get three multiplicity results of infinitely many periodic solutions for system (1.1) using the Fountain theorem, the Dual Fountain theorem, and the Symmetric Mountain Pass theorem.

The rest of the article is divided as follows: Basic definitions and preliminary results are collected in Second 2. The main results and proofs are given in Section 3. The three examples are presented in Section 4 for illustrating our results.

In this article, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M7">View MathML</a> throughout this article, and we use 〈·, ·〉 and |·| to denote the usual inner product and norm in ℝN, respectively.

2. Preliminaries

In this section, we recall some known results in nonsmooth critical point theory, and the properties of space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> are listed for the convenience of readers.

Definition 2.1 [14]. Let p(t) satisfies the condition (A), define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M8">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M9">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M10">View MathML</a>, let u' denote the weak derivative of u, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M11">View MathML</a> and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M12">View MathML</a>

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M13">View MathML</a>

with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M14">View MathML</a>.

In this article, we will use the following equivalent norm on W1, p(t) ([0, T], ℝN), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M15">View MathML</a>

and some lemmas given in the following section have been proven under the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M16">View MathML</a>, and it is obvious that they also hold under the norm ||u||.

Remark 2.1. If p(t) = p, where p ∈ (1, ∞) is a constant, by the definition of |u|p(t), it is easy to get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M17">View MathML</a>, which is the same with the usual norm in space Lp.

The space Lp(t) is a generalized Lebesgue space, and the space W1, p(t) is a generalized Sobolev space. Because most of the following lemmas have appeared in [13,14,17,18], we omit their proofs.

Lemma 2.1 [13]. Lp(t) and W1, p(t) are both Banach spaces with the norms defined above, when p- > 1, they are reflexive.

Lemma 2.2 [14]. (i) The space Lp(t) is a separable, uniform convex Banach space, its conjugate space is Lq(t), for any u Lp(t) and v Lq(t), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M18">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M19">View MathML</a>.

(ii) If p1(t) and p2(t) ∈ C([0, T], ℝ+) and p1(t) ≤ p2(t) for any t ∈ [0, T], then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M20">View MathML</a>, and the embedding is continuous.

Lemma 2.3 [14]. If we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M21">View MathML</a>, ∀ u Lp(t), then

(i) |u|p(t) < 1 (= 1; > 1) ⇔ ρ(u) < 1 (= 1; > 1);

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M22">View MathML</a>;

(iii) |u|p(t) → 0 ⇔ ρ(u) → 0; |u|p(t) → ∞ ⇔ ρ(u) → ∞.

(iv) For u ≠ 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M23">View MathML</a>.

Similar to Lemma 2.3, we have

Lemma 2.4. If we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M24">View MathML</a>, ∀ u W1,p(t), then

(i) ||u|| < 1 (= 1; > 1) ⇔ I(u) < 1 (= 1; > 1);

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M25">View MathML</a>;

(iii) ||u|| → 0 ⇔ I(u) → 0; ||u|| → ∞ ⇔ I(u) → ∞.

(iv) For u ≠ 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M26">View MathML</a>.

Defnition 2.2 [17].

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M27">View MathML</a>

with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M28">View MathML</a>.

For a constant p ∈ (1, ∞), using another conception of weak derivative which is called T-weak derivative, Mawhin and Willem gave the definition of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M6">View MathML</a> by the following way.

Definition 2.3 [17]. Let u L1([0, T], ℝN) and v L1([0, T], ℝN), if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M29">View MathML</a>

then v is called a T-weak derivative of u and is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M30">View MathML</a>.

Definition 2.4 [17]. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M31">View MathML</a>

with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M32">View MathML</a>.

Definition 2.5 [13]. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M33">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M34">View MathML</a> to be the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M35">View MathML</a> in W1,p(t) ([0, T], ℝN).

Remark 2.2. From Definition 2.4, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M36">View MathML</a>, it is easy to conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M37">View MathML</a>.

Lemma 2.5 [13].

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M38">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M39">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M40">View MathML</a>;

(iii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M41">View MathML</a>, then the derivative u' is also the T-weak derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M30">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M42">View MathML</a>.

Lemma 2.6 [17]. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M43">View MathML</a>, then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M44">View MathML</a>,

(ii) u has its continuous representation, which is still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M45">View MathML</a>, u(0) = u(T),

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M30">View MathML</a> is the classical derivative of u, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M46">View MathML</a>.

Since every closed linear subspace of a reflexive Banach space is also reflexive, we have

Lemma 2.7 [13]. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M34">View MathML</a> is a reflexive Banach space if p- > 1.

Obviously, there are continuous embeddings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M48">View MathML</a>. By the classical Sobolev embedding theorem, we obtain

Lemma 2.8 [13]. There is a continuous embedding

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M49">View MathML</a>

when p- > 1, the embedding is compact.

Lemma 2.9 [13]. Each of the following two norms is equivalent to the norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a>:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M50">View MathML</a>, 1 ≤ q ≤ ∞;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M51">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M52">View MathML</a>.

Lemma 2.10 [13]. If u, un Lp(t) (n = 1,2,...), then the following statements are equivalent to each other

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M53">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M54">View MathML</a>;

(iii) un u in measure in [0, T] and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M55">View MathML</a>.

Lemma 2.11 [14]. The functional J defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M56">View MathML</a>

is continuously differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> and J' is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M57">View MathML</a>

(2.1)

and J' is a mapping of (S+), i.e., if un u weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M58">View MathML</a>

then un has a convergent subsequence on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a>.

Lemma 2.12 [18]. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> is a separable and reflexive Banach space, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M60">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M61">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M63">View MathML</a>.

For k = 1, 2,..., denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M64">View MathML</a>

(2.2)

Lemma 2.13 [19]. Let X be a reflexive infinite Banach space, ϕ C1(X, ℝ) is an even functional with the (C) condition and ϕ(0) = 0. If X = Y V with dimY < ∞, and ϕ satisfies

(i) there are constants σ, α > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M65">View MathML</a>,

(ii) for any finite-dimensional subspace W of X, there exists positive constants R2(W) such that ϕ(u) ≤ 0 for u W\Br(0), where Br(0) is an open ball in W of radius r centered at 0. Then ϕ possesses an unbounded sequence of critical values.

Lemma 2.14 [15]. Suppose

(A1) ϕ C1(X, ℝ) is an even functional, then the subspace Xk, Yk, and Zk are defined by (2.2);

If for every k ∈ ℕ, there exists ρk > rk > 0 such that

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M66">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M67">View MathML</a>;

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M68">View MathML</a>, as k → ∞, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M69">View MathML</a>;

(A4) ϕ satisfies the (PS)c condition for every c > 0.

Then ϕ has an unbounded sequence of critical values.

Lemma 2.15 [16]. Assume (A1) is satisfied, and there is a k0 > 0 so as to for each k k0, there exist ρk > rk > 0 such that

(A5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M70">View MathML</a>, as k → ∞;

(A6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M71">View MathML</a>;

(A7) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M72">View MathML</a>;

(A8) ϕ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M73">View MathML</a> condition for every c ∈ [dk0, 0).

Then ϕ has a sequence of negative critical values converging to 0.

Remark 2.3. ϕ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M73">View MathML</a> condition means that if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M74">View MathML</a> such that nj → ∞, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M76">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M77">View MathML</a> contains a subsequence converging to critical point of ϕ. It is obvious that if ϕ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M73">View MathML</a> condition, then ϕ satisfies the (PS)c condition.

3. Main results and proofs of the theorems

Theorem 3.1. Let F(t, x) satisfies the condition (A'), and suppose the following conditions hold:

(B1) there exist β > p+ and r > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M78">View MathML</a>

for a.e. t ∈ [0, T] and all |x| ≥ r in ℝN;

(B2) there exist positive constants μ > p+ and Q > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M79">View MathML</a>

uniformly for a.e. t ∈ [0, T];

(B3) there exists μ' > p+ and Q' > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M80">View MathML</a>

uniformly for a.e. t ∈ [0, T];

(B4) F(t, x) = F(t, -x) for t ∈ [0, T] and all x in ℝN.

Then system (1.1) has infinite solutions uk in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> for every positive integer k such that ||uk||→ +∞, as k → ∞.

Remark 3.1. Suppose that F(t, ·) is continuously differentiable in x and p(t) ≡ p, then condition (B1) reduces to the well-known Ambrosetti-Rabinowitz condition (see [19]), which was introduced in the context of semi-linear elliptic problems. This condition implies that F(t, x) grows at a superquadratic rate as |x| → ∞. This kind of technical condition often appears as necessary to use variational methods when we solve super-linear differential equations such as elliptic problems, Dirac equations, Hamiltonian systems, wave equations, and Schrödinger equations.

Theorem 3.2. Assume that F(t, x) satisfies (A'), (B1), (B3), and (B4) and the following assumption:

(B5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M81">View MathML</a>, and there exists r1 > p+ and M > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M82">View MathML</a>

Then system (1.1) has infinite solutions uk in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> for every positive integer k such that ||uk||→ +∞, as k → ∞.

Theorem 3.3. Assume that F(t, x) satisfies the following assumption:

(B6) F(t, x):= a(t)|x|γ, where a(t) ∈ L(0, T; ℝ+) and 1 < γ < p- is a constant. Then system (1.1) has infinite solutions uk in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> for every positive integer k.

The proof of Theorem 3.1 is organized as follows: first, we show the functional ϕ defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M83">View MathML</a>

satisfies the (PS) condition, then we verify for ϕ the conditions in Lemma 2.14 item-by-item, then ϕ has an unbounded sequence of critical values.

Proof of Theorem 3.1. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M84">View MathML</a> such that ϕ(un) is bounded and ϕ'(un) → 0 as n → ∞. First, we prove {un} is a bounded sequence, otherwise, {un} would be unbounded sequence, passing to a subsequence, still denoted by {un}, such that ||un|| ≥ 1 and ||un|| → ∞. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M85">View MathML</a>

(3.1)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M86">View MathML</a>.

It follows from (3.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M87">View MathML</a>

(3.2)

where Ω1:= {t ∈ [0, T]; |un(t)| ≤ r}, Ω2:= [0, T] \ Ω1 and C0 is a positive constant.

However, from (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M88">View MathML</a>

Thus ||un|| is a bounded sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a>.

By Lemma 2.8, the sequence {un} has a subsequence, also denoted by {un}, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M89">View MathML</a>

(3.3)

and ||u||C1||u|| by Lemma 2.8, where C1 is a positive constant.

Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M90">View MathML</a>

(3.4)

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M91">View MathML</a>

(3.5)

By (3.4) and (3.5), we get 〈J'(u) - J'(un), u - un〉 → 0, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M92">View MathML</a>

so it follows Lemma 2.11 that {un} admits a convergent subsequence.

For any u Yk, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M93">View MathML</a>

(3.6)

and it is easy to verify that ||·||* defined by (3.6) is a norm of Yk. Since all the norms of a finite dimensional normed space are equivalent, so there exists positive constant C2 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M94">View MathML</a>

(3.7)

In view of (B3), there exist two positive constants M1 and C3 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M95">View MathML</a>

(3.8)

for a.e. t ∈ [0, T] and |x| ≥ C3.

It follows (3.7) and (3.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M96">View MathML</a>

where Ω3:= {t ∈ [0, T]; |u(t)| ≥ C3}, Ω4:= [0, T] \ Ω3 and C4 is a positive constant.

Since μ' > p+, there exist positive constants dk such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M97">View MathML</a>

(3.9)

For any u Zk, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M98">View MathML</a>

(3.10)

then we conclude βk → 0 as k → ∞.

In fact, it is obvious that βk βk + 1 > 0, so βk β ≥ 0 as k → ∞. For every k ∈ ℕ, there exists uk Zk such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M99">View MathML</a>

(3.11)

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> is reflexive, {uk} has a weakly convergent subsequence, still denoted by {uk}, such that uk u. We claim u = 0.

In fact, for any fm ∈ {fn: n = 1, 2...,}, we have fm(uk) = 0, when k > m, so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M100">View MathML</a>

for any fm ∈ {fn: n = 1, 2 ...,}, therefore u = 0.

By Lemma 2.8, when uk ⇀ 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a>, then uk → 0 strongly in C([0, T]; ℝN). So, we conclude β = 0 by (3.11).

In view of (B2), there exist two positive constants M2 and C10 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M101">View MathML</a>

(3.12)

uniformly for a.e. t ∈ [0, T] and |x| ≥ C5.

When ||u|| ≥ 1, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M102">View MathML</a>

where Ω5:= {t ∈ [0, T]; |u(t)| ≥ C5}, Ω6:= [0, T] \ Ω5 and C6 is a positive constant.

Choosing rk = 1/βk, it is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M103">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M104">View MathML</a>

(3.13)

i.e., the condition (A3) in Lemma 2.14 is satisfied.

In view of (3.9), let ρk:= max{dk, rk + 1}, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M105">View MathML</a>

and this shows the condition of (A2) in Lemma 2.14 is satisfied.

We have proved the functional ϕ satisfies all the conditions of Lemma 2.14, then ϕ has an unbounded sequence of critical values ck = ϕ(uk) by Lemma 2.14, we only need to show ||uk||→ ∞ as k → ∞.

In fact, since uk is a critical point of the functional ϕ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M106">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M107">View MathML</a>

(3.14)

since ck → ∞, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M108">View MathML</a>

by (3.14). In fact, if not, going to a subsequence if necessary, we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M109">View MathML</a>

for all k ∈ ℕ and some positive constant C7.

Combining (A') and (3.14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M110">View MathML</a>

which contradicts ck → ∞. This completes the proof of Theorem 3.1.

Proof of Theorem 3.2. To prove {un} has a convergent subsequence in space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> is the same as that in the proof of Theorem 3.1, thus we omit it. It is obvious that ϕ is even and ϕ(0) = 0 under condition (B5), and so we only need to verify other conditions in Lemma 2.13.

Proposition 3.1. Under the condition (B5), there exist two positive constants σ and α such that ϕ(u) ≥ α for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M111">View MathML</a> and ||u|| = σ.

Proof. In view of condition (B5), there exist two positive constants ε and δ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M112">View MathML</a>

where C1 is the same as in (3.3), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M113">View MathML</a>

(3.15)

for a.e. t ∈ [0, T] and |x| ≤ δ.

Let σ:= δ/C1 and ||u|| = σ, since σ < 1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M114">View MathML</a>

(3.16)

by Lemmas 2.4 and 2.8.

Combining (3.15) and (3.16), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M115">View MathML</a>

so we can choose σ small enough, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M116">View MathML</a>

and this completes the proof of Proposition 3.1.

Proposition 3.2. For any finite dimensional subspace W of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a>, there is r2 = r2(W) > 0 such that ϕ(u) ≤ 0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M117">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M118">View MathML</a> is an open ball in W of radius r2 centered at 0.

Proof. The proof of Proposition 3.2 is the same as the proof of the condition (A2) in the proof of Theorem 3.1.

We have proved the functional ϕ satisfies all the conditions of Lemma 2.13, ϕ has an unbounded sequence of critical values ck = ϕ(uk) by Lemma 2.13. Arguing as in the proof of Theorem 3.1, system (1.1) has infinite solutions {uk} in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> for every positive integer k such that ||uk||→ +∞, as k → ∞. The proof of Theorem 3.2 is complete.

Proof of Theorem 3.3. First, we show that ϕ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M73">View MathML</a> for every c ∈ ℝ. Suppose nj → ∞, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M76">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M77">View MathML</a> is a bounded sequence, otherwise, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M77">View MathML</a> would be unbounded sequence, passing to a subsequence, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M77">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M119">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M120">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M121">View MathML</a>

(3.17)

However, from (3.17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M122">View MathML</a>

thus ||un|| is a bounded sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a>. Going, if necessary, to a subsequence, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M123">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M124">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M125">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M126">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M127">View MathML</a>

In view of (3.4) and (3.5), we can also conclude <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M128">View MathML</a>, furthermore, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M129">View MathML</a>.

Let us prove ϕ'(u) = 0 below. Taking arbitrarily ωk Yk, notice when nj k we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M130">View MathML</a>

Going to limit in the right side of above equation reaches

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M131">View MathML</a>

so ϕ'(u) = 0, this shows that ϕ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M73">View MathML</a> for every c ∈ ℝ.

For any finite dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M132">View MathML</a>, there exists ε1 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M133">View MathML</a>

(3.18)

Otherwise, for any positive integer n, there exists un W \ {0} such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M134">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M135">View MathML</a>, then ||vn|| = 1 for all n ∈ ℕ and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M136">View MathML</a>

(3.19)

Since dimW < ∞, it follows from the compactness of the unit sphere of W that there exists a subsequence, denoted also by {vn}, such that {vn} converges to some v0 in W. It is obvious that ||v0|| = 1.

By the equivalence of the norms on the finite dimensional space W, we have vn v0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M137">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M138">View MathML</a>

(3.20)

By (3.20) and Hölder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M139">View MathML</a>

(3.21)

Thus, there exist ξ1, ξ2 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M140">View MathML</a>

(3.22)

In fact, if not, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M141">View MathML</a>

for all positive integer n.

It implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M142">View MathML</a>

as n → ∞, where C6 is the same in (3.3). Hence v0 = 0 which contradicts that ||v0|| = 1. Therefore, (3.22) holds. Now let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M143">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M144">View MathML</a>.

By (3.19) and (3.22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M145">View MathML</a>

for all positive integer n. Let n be large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M146">View MathML</a>

then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M147">View MathML</a>

for all large n, which is a contradiction to (3.21). Therefore, (3.18) holds.

For any u Zk, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M148">View MathML</a>

then we conclude γk → 0 as k → ∞ as in the proof of Theorem 3.1.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M149">View MathML</a>

(3.23)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M150">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M151">View MathML</a>, it is obvious that ρk → 0, as k → ∞. In view of (3.23), We conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M152">View MathML</a>

so the condition (A7) in Lemma 2.15 is satisfied.

Furthermore, by (3.23), for any u Zk with ||u|| ≤ ρk, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M153">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M154">View MathML</a>

so we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M155">View MathML</a>

for ρk, γk → 0, as k → ∞.

For any u Yk \ {0} with ||u|| ≤ 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M156">View MathML</a>

where ε1 is given in (3.18), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M157">View MathML</a>

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M158">View MathML</a>, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M159">View MathML</a>

i.e., the condition (A6) in Lemma 2.15 is satisfied. The proof of Theorem 3.3 is complete.

4. Example

In this section, we give three examples to illustrate our results.

Example 4.1. In system (1.1), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M160">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M161">View MathML</a>

Choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M162">View MathML</a>

so it is easy to verify that all the conditions (B1)-(B4) are satisfied. Then by Theorem 3.1, system (1.1) has infinite solutions {uk} in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> for every positive integer k such that ||uk||→ +∞, as k → ∞.

Example 4.2. In system (1.1), let F(t, x) = |x|8 and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M163">View MathML</a>

We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M164">View MathML</a>, r = 2, μ' = 8, r1 = 7, Q' = 1 and M = 1, so it is easy to verify that all the conditions of Theorem 3.2 are satisfied. Then by Theorem 3.2, so system (1.1) has infinite solutions {uk} in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> for every positive integer k such that ||uk||→ +∞, as k → ∞.

Example 4.3. In system (1.1), let F(t, x) = a(t)|x|3 where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M165">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M166">View MathML</a>

It is easy to verify that all the conditions of Theorem 3.3 are satisfied. Then by Theorem 3.3, so system (1.1) has infinite solutions {uk} in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/33/mathml/M5">View MathML</a> for every positive integer k.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All the authors typed, read, and approved the final manuscript.

Acknowledgements

The authors thank the anonymous referees for valuable suggestions and comments which led to improve this article. This Project is supported by the National Natural Science Foundation of China (Grant No. 11171351).

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