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

Li Yin; Jinghua Yao; Qihu Zhang; Chunshan Zhao

doi:10.3934/dcds.2017095

Article Contents

2017, Volume 37, Issue 4: 2207-2226. Doi: 10.3934/dcds.2017095

This issue Previous Article Global attractor for a strongly damped wave equation with fully supercritical nonlinearities Next Article Suspension of the billiard maps in the Lazutkin's coordinate

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
*Corresponding author: Jinghua Yao

Received: June 30, 2016

Revised: August 31, 2016

Published: May 2017

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

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Abstract

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.

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Mathematics Subject Classification: Primary:35J20, 35J25;Secondary:35J60.

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