- Research Article
- Open Access
Infinitely Many Periodic Solutions for Variable Exponent Systems
© Xiaoli Guo et al. 2009
- Received: 4 December 2008
- Accepted: 14 April 2009
- Published: 1 June 2009
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.
- Differential Equation
- Growth Condition
- Periodic Solution
- Bounded Domain
- Variational Problem
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.
the authors give the existence of positive solutions for problem .
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 ). 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 .
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.
Write , , for any . Assume that
For any positive constant , there are .
, and are odd functions such that , and are even, and is a periodic of and , namely,
Note. In , 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.
If hold, then possesses a positive solution, when is sufficiently large.
which satisfy and , then possesses a positive solution (see ).
We will construct a subsolution of .
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,
Then is a subsolution of .
We will construct a supersolution of .
where is a positive constant and .
According to (2.24) and (2.26), we can conclude that is a supersolution for , when is large enough.
We will prove that and .
According to the comparison principle (see ), we can see that .
According to the comparison principle (see ), we can see that .
Thus, we can conclude that and , when is sufficiently large. This completes the proof.
If hold, then has infinitely many periodic solutions.
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.
Partly supported by the National Science Foundation of China (10701066 & 10671084) and China Postdoctoral Science Foundation (20070421107) and the Natural Science Foundation of Henan Education Committee (2008-755-65 & 2009A120005) and the Natural Science Foundation of Jiangsu Education Committee (08KJD110007).
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