    April  2017, 37(4): 2207-2226. doi: 10.3934/dcds.2017095

Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth

 1 College of Information and Management Science, Henan Agricultural University, Zhengzhou, Henan 450002, China 2 Department of Mathematics, Indiana University, Bloomington, IN 47408, USA 3 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China 4 Department of Mathematical Science, Georgia Southern University, Statesboro, GA 30460, USA

*Corresponding author: Jinghua Yao

Received  July 2016 Revised  September 2016 Published  December 2016

Fund Project: This research is partly supported by the key projects in Science and Technology Research of the Henan Education Department (14A110011).

We investigate the followingDirichlet problem with variable exponents:
 \left\{ \begin{align} &-{{\vartriangle }_{p(x)}}u=\lambda \alpha (x)|u{{\text{ }\!\!|\!\!\text{ }}^{\alpha (x)-2}}u|v{{\text{ }\!\!|\!\!\text{ }}^{\beta (x)}}+{{F}_{u}}(x,u,v),\text{ in }\Omega , \\ &-{{\vartriangle }_{q(x)}}v=\lambda \beta (x)\text{ }\!\!|\!\!\text{ }u{{|}^{\alpha \left( x \right)}}|v{{|}^{\beta (x)\text{-2}}}v+{{F}_{v}}(x,u,v),\text{ in}\ \Omega , \\ &u=0=v,\text{ on }\partial \Omega \text{.} \\ \end{align} \right.
We present here, in the system setting, a new set of growth conditions under which we manage to use a novel method to verify the Cerami compactness condition. By localization argument, decomposition technique and variational methods, we are able to show the existence of multiple solutions with constant sign for the problem without the well-knownAmbrosetti-Rabinowitz type growth condition. More precisely, we manage to show that the problem admitsfour, six and infinitely many solutions respectively.
Citation: Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao. Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095
References:
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References:
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