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

In this article, we study the existence of solution for the following elliptic system of variable exponents with perturbation terms
 -\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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