Infinitely Many Periodic Solutions for Variable Exponent Systems

Abstract

We mainly consider the system in , in , where are periodic functions, and is called -Laplacian. We give the existence of infinitely many periodic solutions under some conditions.

1. Introduction

The study of differential equations and variational problems with variable exponent growth conditions has been a new and interesting topic. Many results have been obtained on this kind of problems, for example [1–18]. On the applied background, we refer to [1, 3, 11, 18]. In this paper, we mainly consider the existence of infinitely many periodic solutions for the system

(1.1)

where are functions. The operator is called one-dimensional -Laplacian. Especially, if (a constant) and (a constant), then is the well-known constant exponent system.

is called a solution of , if , and are absolute continuous and satisfy almost every where.

In [19], the authors consider the existence of positive weak solutions for the following constant exponent problems:

(1.2)

The first eigenfunction is used to construct the subsolution of constant exponent problems successfully. Under the condition that is large enough and

(1.3)

the authors give the existence of positive solutions for problem .

In [20], the author considers the existence and nonexistence of positive weak solution to the following constant exponent elliptic system:

(1.4)

The first eigenfunction is used to construct the subsolution of constant exponent problems successfully.

Because of the nonhomogeneity of -Laplacian, -Laplacian problems are more complicated than those of -Laplacian. Maybe the first eigenvalue and the first eigenfunction of -Laplacian do not exist (see [6]). Even if the first eigenfunction of -Laplacian exists, because of the nonhomogeneity of -Laplacian, the first eigenfunction cannot be used to construct the subsolution of -Laplacian problems.

There are many papers on the existence of periodic solutions for -Laplacian elliptic systems, for example [21–24]. The results on the periodic solutions for variable exponent systems are rare. Through a new method of constructing sub-supersolution, this paper gives the existence of infinitely many periodic solutions for problem .

2. Main Results and Proofs

At first, we give an existence of positive solutions for variable exponent systems on bounded domain via sub-super-solution method. The result itself has dependent value.

Denote . Let us consider the existence of positive solutions of the following:

(2.1)

Write , , for any . Assume that

satisfy

(2.2)

, are , monotone functions such that

(2.3)

For any positive constant , there are .

.

, and are odd functions such that , and are even, and is a periodic of and , namely,

Note. In [14], the present author discussed the existence of solutions of , under the conditions that is radial, , and . Because of the nonhomogeneity of variable exponent problems, variable exponent problems are more complicated than constant exponent problems, and many results and methods for constant exponent problems are invalid for variable exponent problems. In many cases, the radial symmetric conditions are effective to deal with variable exponent problems. There are many results about the radial variable exponent problems (see [4, 14, 16]), but the following Theorem 2.1 does not assume any symmetric conditions.

We will establish.

Theorem 2.1.

If hold, then possesses a positive solution, when is sufficiently large.

Proof.

If we can construct a positive subsolution and supersolution of , namely,

(2.4)

which satisfy and , then possesses a positive solution (see [5]).

Step 1.

We will construct a subsolution of .

Denfine

(2.5)

where

(2.6)

and satisfy

(2.7)

and satisfy

(2.8)

then , and . It is easy to see that and , . Obviously, is continuous about .

In the following, we will prove that is a subsolution for . By computation,

(2.9)

If is sufficiently large, we have

(2.10)

As is a constant and only depends on and , when is large enough, we have . Since and is monotone, we have

(2.11)

According to (), when are large enough, we have

(2.12)

where are dependent on , , , and , and they are independent on . Since , when , we have

(2.13)

Then we have

(2.14)

Obviously

(2.15)

When is large enough, from (2.7) we can see that is large enough. Similar to the discussion of the above, we can conclude

(2.16)

(2.17)

Since combining (2.11), (2.14), (2.15), (2.16) and (2.17), we have

(2.18)

Similarly, when is large enough, we have

(2.19)

Then is a subsolution of .

Step 2.

We will construct a supersolution of .

Let be a solution of

(2.20)

where is a positive constant and .

Obviously, there exists such that . Note that is dependent on . Denote . It is easy to see that

(2.21)

Let us consider

(2.22)

Similarly, we have

(2.23)

We will prove that is a supersolution for . From and (2.23), when is large enough, we can easily see that

(2.24)

Since and , when is large enough, according to (2.21) and (2.23), we have

(2.25)

This means that

(2.26)

According to (2.24) and (2.26), we can conclude that is a supersolution for , when is large enough.

Step 3.

We will prove that and .

Obviously, when is large enough, we can easily see that is large enough, then

(2.27)

Let us consider

(2.28)

It is easy to see that is a subsolution of (2.28), when is large enough. Obviously, we can see that is a supersolution of (2.28), and

(2.29)

According to the comparison principle (see [12]), we can see that .

Let us consider

(2.30)

It is easy to see that is a subsolution of (2.30), when is large enough. Obviously, we can see that is a supersolution of (2.30), and

(2.31)

According to the comparison principle (see [12]), we can see that .

Thus, we can conclude that and , when is sufficiently large. This completes the proof.

Theorem 2.2.

If hold, then has infinitely many periodic solutions.

Proof.

Let . According to Theorem 2.1, we can conclude that there exists an integer which is large enough such that has a positive solution for any integer . Since and are even, and are odd, then is a negative solution of . We can define a function on as

(2.32)

We extend as where is an integer such that . It is easy to see that is a solution of and the periodic of is . This completes the proof.