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Periodic solutions for nonautonomous second order Hamiltonian systems with sublinear nonlinearity

Zhiyong Wang1* and Jihui Zhang2

Author Affiliations

1 Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, Jiangsu, People's Republic of China

2 Jiangsu Key Laboratory for NSLSCS, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210097, Jiangsu, People's Republic of China

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Boundary Value Problems 2011, 2011:23  doi:10.1186/1687-2770-2011-23

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/23


Received:14 May 2011
Accepted:13 September 2011
Published:13 September 2011

© 2011 Wang and Zhang; 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

Some existence and multiplicity of periodic solutions are obtained for nonautonomous second order Hamiltonian systems with sublinear nonlinearity by using the least action principle and minimax methods in critical point theory.

Mathematics Subject Classification (2000): 34C25, 37J45, 58E50.

Keywords:
Control function; Periodic solutions; The least action principle; Minimax methods

1 Introduction and main results

Consider the second order systems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M1">View MathML</a>

(1.1)

where T > 0 and F : [0, T] × ℝ→ ℝ satisfies the following assumption:

(A) F (t, x) is measurable in t for every x ∈ ℝand 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/2011/1/23/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M2">View MathML</a>

for all x ∈ ℝand a.e. t ∈ [0, T].

The existence of periodic solutions for problem (1.1) has been studied extensively, a lot of existence and multiplicity results have been obtained, we refer the readers to [1-13] and the reference therein. In particular, under the assumptions that the nonlinearity ∇F (t, x) is bounded, that is, there exists p(t) ∈ L1(0, T ; ℝ+) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M3">View MathML</a>

(1.2)

for all x ∈ ℝand a.e. t ∈ [0, T], and that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M4">View MathML</a>

(1.3)

Mawhin and Willem in [3] have proved that problem (1.1) admitted a periodic solution. After that, when the nonlinearity ∇F (t, x) is sublinear, that is, there exists f(t), g(t) ∈ L1(0, T ; ℝ+) and α ∈ [0, 1) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M5">View MathML</a>

(1.4)

for all x ∈ ℝand a.e. t ∈ [0, T], Tang in [7] have generalized the above results under the hypotheses

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M6">View MathML</a>

(1.5)

Subsequently, Meng and Tang in [13] further improved condition (1.5) with α ∈ (0, 1) by using the following assumptions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M7">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M8">View MathML</a>

(1.7)

Recently, authors in [14] investigated the existence of periodic solutions for the second order nonautonomous Hamiltonian systems with p-Laplacian, here p > 1, it is assumed that the nonlinearity ∇F (t, x) may grow slightly slower than |x|p-1, a typical example with p = 2 is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M9">View MathML</a>

(1.8)

solutions are found as saddle points to the corresponding action functional. Furthermore, authors in [12] have extended the ideas of [14], replacing in assumptions (1.4) and (1.5) the term |x| with a more general function h(|x|), which generalized the results of [3,7,10,11]. Concretely speaking, it is assumed that there exist f(t), g(t) ∈ L1(0, T; ℝ+) and a nonnegative function h C([0, +∞), [0, +∞)) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M10">View MathML</a>

for all x ∈ ℝand a.e. t ∈ [0, T], and that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M11">View MathML</a>

where h be a control function with the properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M12">View MathML</a>

if α = 0, h(t) only need to satisfy conditions (a)-(c), here C*, K1 and K2 are positive constants. Moreover, α ∈ [0, 1) is posed. Under these assumptions, periodic solutions of problem (1.1) are obtained. In addition, if the nonlinearity ∇F (t, x) grows more faster at infinity with the rate like <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M13">View MathML</a>, f(t) satisfies some certain restrictions and α is required in a more wider range, say, α ∈ [0,1], periodic solutions have also been established in [12] by minimax methods.

An interesting question naturally arises: Is it possible to handle both the case such as (1.8) and some cases like (1.4), (1.5), in which only f(t) ∈ L1(0, T ; ℝ+) and α ∈ [0, 1)? In this paper, we will focus on this problem.

We now state our main results.

Theorem 1.1. Suppose that F satisfies assumption (A) and the following conditions:

(S1) There exist constants C ≥ 0, C* > 0 and a positive function h C(ℝ+, ℝ+) with the properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M14">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M15">View MathML</a>. Moreover, there exist f L1(0, T; ℝ+) and g L1(0, T; ℝ+) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M16">View MathML</a>

for all x ∈ ℝand a.e. t ∈ [0, T];

(S2) There exists a positive function h C(ℝ+, ℝ+) which satisfies the conditions (i)-(iv) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M17">View MathML</a>

Then, problem (1.1) has at least one solution which minimizes the functional φ given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M18">View MathML</a>

on the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M20">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M21">View MathML</a>

Theorem 1.2. Suppose that (S1) and assumption (A) hold. Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M22">View MathML</a>

Then, problem (1.1) has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>.

Theorem 1.3. Suppose that (S1), (S3) and assumption (A) hold. Assume that there exist δ > 0, ε > 0 and an integer k > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M23">View MathML</a>

(1.9)

for all x ∈ ℝand a.e. t ∈ [0, T], and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M24">View MathML</a>

(1.10)

for all |x| ≤ δ and a.e. t ∈ [0, T], where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M25">View MathML</a>. Then, problem (1.1) has at least two distinct solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>.

Theorem 1.4. Suppose that (S1), (S2) and assumption (A) hold. Assume that there exist δ > 0, ε > 0 and an integer k ≥ 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M26">View MathML</a>

(1.11)

for all |x| ≤ δ and a.e. t ∈ [0, T]. Then, problem (1.1) has at least three distinct solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>.

Remark 1.1.

(i) Let α ∈ [0, 1), in Theorems 1.1-1.4, ∇F(t, x) does not need to be controlled by |x|2α at infinity; in particular, we can not only deal with the case in which ∇F(t, x) grows slightly faster than |x|2α at infinity, such as the example (1.8), but also we can treat the cases like (1.4), (1.5).

(ii) Compared with [12], we remove the restriction on the function f(t) as well as the restriction on the range of α ∈ [0, 1] when we are concerned with the cases like (1.8).

(iii) Here, we point out that introducing the control function h(t) has also been used in [12,14], however, these control functions are different from ours because of the distinct characters of h(t).

Remark 1.2. From (i) of (S1), we see that, nonincreasing control functions h(t) can be permitted. With respect to the detailed example on this assertion, one can see Example 4.3 of Section 4.

Remark 1.3. There are functions F(t, x) satisfying our theorems and not satisfying the results in [1-14]. For example, consider function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M27">View MathML</a>

where f(t) ∈ L1(0, T; ℝ+) and f(t) > 0 for a.e. t ∈ [0, T]. It is apparent that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M28">View MathML</a>

(1.12)

for all x ∈ ℝand t ∈ [0, T]. (1.12) shows that (1.4) does not hold for any α ∈ [0, 1), moreover, note f(t) only belongs to L1(0, T; ℝ+) and no further requirements on the upper bound of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M29">View MathML</a> are posed, then the approach of [12] cannot be repeated. This example cannot be solved by earlier results, such as [1-13].

On the other hand, take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M31">View MathML</a>, C = 0, C* = 1, then by simple computation, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M32">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M33">View MathML</a>

Hence, (S1) and (S2) are hold, by Theorem 1.1, problem (1.1) has at least one solution which minimizes the functional φ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>.

What's more, Theorem 1.1 can also deal with some cases which satisfy the conditions (1.4) and (1.5). For instance, consider function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M34">View MathML</a>

where q(t) ∈ L1(0, T; ℝ). It is not difficult to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M35">View MathML</a>

for all x ∈ ℝand a.e. t ∈ [0, T]. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M37">View MathML</a>, C = 0, C* = 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M38">View MathML</a> and g(t) = |q(t)|, then (S1) and (S2) hold, by Theorem 1.1, problem (1.1) has at least one solution which minimizes the functional φ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>. However, we can find that the results of [14] cannot cover this case. More examples are drawn in Section 4.

Our paper is organized as follows. In Section 2, we collect some notations and give a result regrading properties of control function h(t). In Section 3, we are devote to the proofs of main theorems. Finally, we will give some examples to illustrate our results in Section 4.

2 Preliminaries

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M39">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M41">View MathML</a>, then one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M42">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M43">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M44">View MathML</a>.

It follows from assumption (A) that the corresponding function φ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M45">View MathML</a>

is continuously differentiable and weakly lower semi-continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a> (see[2]). Moreover, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M46">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M47">View MathML</a>. It is well known that the solutions of problem (1.1) correspond to the critical point of φ.

In order to prove our main theorems, we prepare the following auxiliary result, which will be used frequently later on.

Lemma 2.1. Suppose that there exists a positive function h which satisfies the conditions (i), (iii), (iv) of (S1), then we have the following estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M48">View MathML</a>

Proof. It follows from (iv) of (S1) that, for any ε > 0, there exists M1 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M49">View MathML</a>

(2.1)

By (iii) of (S1), there exists M2 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M50">View MathML</a>

(2.2)

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M51">View MathML</a>

(2.3)

where M := max{M1, M2}. Hence, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M52">View MathML</a>

(2.4)

for all t > 0 by (i) of (S1). Obviously, h(t) satisfies (1) due to the definition of h(t) and (2.4).

Next, we come to check condition (2). Recalling the property (iv) of (S1) and (2.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M53">View MathML</a>

Therefore, condition (2) holds.

Finally, we show that (3) is also true. By (iii) of (S1), one arrives at, for every β > 0, there exists M3 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M54">View MathML</a>

(2.5)

Let θ ≥ 1, using (2.5) and integrating the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M55">View MathML</a>

over an interval [1, S] ⊂ [1, +∞), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M56">View MathML</a>

Thus, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M57">View MathML</a> by (iv) of (S1), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M58">View MathML</a>

for all t M3. That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M59">View MathML</a>

which completes the proof. □

3 Proof of main results

For the sake of convenience, we will denote various positive constants as Ci, i = 1, 2, 3,.... Now, we are ready to proof our main results.

Proof of Theorem 1.1. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M60">View MathML</a>, it follows from (S1), Lemma 2.1 and Sobolev's inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M61">View MathML</a>

(3.1)

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M62">View MathML</a>

(3.2)

Taking into account Lemma 2.1 and (S2), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M63">View MathML</a>

(3.3)

As ||u|| → +∞ if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M64">View MathML</a>, for ε small enough, (3.2) and (3.3) deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M65">View MathML</a>

Hence, by the least action principle, problem (1.1) has at least one solution which minimizes the function φ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>. □

Proof of Theorem 1.2. First, we prove that φ satisfies the (PS) condition. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M66">View MathML</a> is a (PS) sequence of φ, that is, φ'(un) → 0 as n → +∞ and {φ(un)} is bounded. In a way similar to the proof of Theorem 1.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M67">View MathML</a>

for all n. Hence, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M68">View MathML</a>

(3.4)

for large n. On the other hand, it follows from Wirtinger's inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M69">View MathML</a>

(3.5)

for all n. Combining (3.4) with (3.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M70">View MathML</a>

(3.6)

for all large n. By (3.1), (3.6), Lemma 2.1 and (S3), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M71">View MathML</a>

This contradicts the boundedness of {φ(un)}. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M72">View MathML</a> is bounded. Notice (3.6) and (1) of Lemma 2.1, hence {un} is bounded. Arguing then as in Proposition 4.1 in [3], we conclude that the (PS) condition is satisfied.

In order to apply the saddle point theorem in [2,3], we only need to verify the following conditions:

(φ1) φ(u) → +∞ as ||u|| → +∞in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M73">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M74">View MathML</a> ,

(φ2) φ(u) → -∞ as |u(t)| → +∞.

In fact, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M75">View MathML</a>, by (S1), Sobolev's inequality and Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M76">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M77">View MathML</a>

(3.7)

By Wirtinger's inequality, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M78">View MathML</a>

Hence, for ε small enough, (φ1) follows from (3.7).

On the other hand, by (S3) and Lemma 2.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M79">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M80">View MathML</a>

Thus, (φ2) is verified. The proof of Theorem 1.2 is completed. □

Proof of Theorem 1.3. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M81">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M82">View MathML</a>

and ψ = -φ. Then, ψ C1(E, ℝ) satisfies the (PS) condition by the proof of Theorem 1.2. In view of Theorem 5.29 and Example 5.26 in [2], we only need to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M83">View MathML</a>

We see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M84">View MathML</a>

for all x ∈ ℝand a.e. t ∈ [0, T]. By (S1) and Lemma 2.1, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M85">View MathML</a>

for all |x| ≥ δ, a.e. t ∈ [0, T] and some Q(t) ∈ L1(0, T; ℝ+) given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M86">View MathML</a>

Now, it follows from (1.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M87">View MathML</a>

for all x ∈ ℝand a.e. t ∈ [0, T]. Hence, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M88">View MathML</a>

for all u Hk. Then, (ψ1) follows from the above inequality.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M89">View MathML</a>, by (1.9), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M90">View MathML</a>

So, (ψ2) is obtained. At last, (ψ3) follows from (φ1) which are appeared in the proof of Theorem 1.2. Then the proof of Theorem 1.3 is completed. □

Proof of Theorem 1.4. From the proof of Theorem 1.1, we know that φ is coercive which implies that φ satisfies the (PS) condition. With the similar manner to [4,7], we can get the multiplicity results, here we omit the details. □

4 Examples

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

Example 4.1. Consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M91">View MathML</a>

where d(t) ∈ L1(0, T; ℝ). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M30">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M31">View MathML</a>, by a direct computation, (S1) and (S3) hold. Then, by Theorem 1.2, we conclude that problem (1.1) has one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>. However, as the reason of Remark 1.3, the results in [1-13] cannot be applied.

Example 4.2. Consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M92">View MathML</a>

where A(t), B(t) are suitable functions which insure assumption (A) hold. Also, put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M31">View MathML</a>, we see that (S1), (S2) and (1.11) hold. By virtue of Theorem 1.4, problem (1.1) has at least three distinct solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>.

Example 4.3. Consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M93">View MathML</a>

We observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M94">View MathML</a>

which means ∇F(t, x) is bounded, moreover, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M95">View MathML</a>

Then, by the results in [3,7,12], problem (1.1) has one solution which minimizes the functional φ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>.

In fact, our Theorem 1.1 can also handle this case. In this situation, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M97">View MathML</a>, and choose C = 2, C* = 1 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M98">View MathML</a>, g(t) ≡ 0, we infer

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M99">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M100">View MathML</a>

So, by Theorem 1.1, problem (1.1) has one solution which minimizes the functional φ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>.

Remark 4.1. Unlike the control functions in [12], where h(t) is nondecreasing, here control function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M96">View MathML</a> is bounded but not increasing.

Example 4.4. Consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M101">View MathML</a>

where k(t) ∈ L1(0, T; ℝ). It is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M102">View MathML</a>

The above inequality leads to (1.4) hold with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M103">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M104">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M105">View MathML</a>

So, by the theorems in [3,7,12,13], problem (1.1) has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>.

Indeed, our Theorem 1.2 can also deal with this case. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M107">View MathML</a>, and choose C = 0, C* = 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M108">View MathML</a>, g(t) = |k(t)|, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M109">View MathML</a>

Furthermore, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M110">View MathML</a>

Hence, (S1) and (S3) are true, by Theorem 1.2, problem (1.1) has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/23/mathml/M19">View MathML</a>. However, we can find that the results in [14] cannot deal with this case.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

The authors would like to thank Professor Huicheng Yin for his help and many valuable discussions, and the first author takes the opportunity to thank Professor Xiangsheng Xu and the members at Department of Mathematics and Statistics at Mississippi State University for their warm hospitality and kindness. This Project is Supported by National Natural Science Foundation of China (Grant No. 11026213, 10871096), Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No. 10KJB110006) and Foundation of Nanjing University of Information Science and Technology (Grant No. 20080280).

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