Electron. J. Diff. Equ., Vol. 2015 (2015), No. 63, pp. 1-23.

Existence of solutions for a variable exponent system without PS conditions

Li Yin, Yuan Liang, Qihu Zhang, Chunshan Zhao

Abstract:
In this article, we study the existence of solution for the following elliptic system of variable exponents with perturbation terms
$$\displaylines{ -\hbox{div}| \nabla u| ^{p(x)-2}\nabla u)+|u| ^{p(x)-2}u =\lambda a(x)| u| ^{\gamma(x)-2}u+F_{u}(x,u,v)\quad\hbox{in } \mathbb{R}^N, \cr -\hbox{div}| \nabla v| ^{q(x)-2}\nabla v)+|v| ^{q(x)-2}v =\lambda b(x)| v| ^{\delta(x)-2}v+F_{v}(x,u,v)\quad \hbox{in }\mathbb{R}^N, \cr u\in W^{1,p(\cdot )}(\mathbb{R}^N),v\in W^{1,q(\cdot )}(\mathbb{R}^N), }$$
where the corresponding functional does not satisfy PS conditions. We obtain a sufficient condition for the existence of solution and also present a result on asymptotic behavior of solutions at infinity.

Submitted September 29, 2014. Published March 13, 2015.
Math Subject Classifications: 35J47.
Key Words: Variable exponent system; integral functional; PS condition; variable exponent Sobolev space.

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Li Yin
College of Information and Management Science
Henan Agricultural University
Zhengzhou, Henan 450002, China
email: mathsr@163.com
Yuan Liang
Junior College, Zhejiang Wanli University
Ningbo, Zhejiang 315100, China
email: ly0432@163.com
Qihu Zhang
College of Mathematics and Information Science
Zhengzhou University of Light Industry
Zhengzhou, Henan 450002, China
email: zhangqihu@yahoo.com, zhangqh1999@yahoo.com.cn
Chunshan Zhao
Department of Mathematical Sciences
Georgia Southern University
Statesboro, GA 30460, USA
email: czhao@GeorgiaSouthern.edu

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