- 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
- Differential Equation
- Growth Condition
- Periodic Solution
- Bounded Domain
- Variational 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.
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.
which satisfy and , then possesses a positive solution (see ).
According to the comparison principle (see ), we can see that .
According to the comparison principle (see ), we can see that .
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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