Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces

Volume 10, Issue 7, pp 3792--3814 http://dx.doi.org/10.22436/jnsa.010.07.34

Publication Date: July 26, 2017 Submission Date: April 25, 2017

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Authors

Liben Wang - Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China. - Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China. Xingyong Zhang - Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China. Hui Fang - Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China. - Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China.

Abstract

In this paper, we investigate the following nonlinear and non-homogeneous elliptic system \[ \begin{cases} -div(\phi_1(|\nabla u|)\nabla u)= F_u(x,u,v),\,\,\,\,\, \texttt{in} \Omega,\\ -div(\phi_2(|\nabla v|)\nabla v)= F_v(x,u,v),\,\,\,\,\, \texttt{in} \Omega,\\ u=v=0,\,\,\,\,\, \texttt{on} \partial \Omega. \end{cases} \] where \(\Omega\) is a bounded domain in \(R^N(N \geq 2)\) with smooth boundary \(\partial\Omega\) , functions \(\phi_i(t)t (i = 1, 2)\) are increasing homeomorphisms from \(R^+\) onto \(R^+\). When \(F\) satisfies some \((\phi_1,\phi_2)\)-superlinear and subcritical growth conditions at infinity, by using the mountain pass theorem we obtain that system has a nontrivial solution, and when \(F\) satisfies an additional symmetric condition, by using the symmetric mountain pass theorem, we obtain that system has infinitely many solutions. Some of our results extend and improve those corresponding results in Carvalho et al. [M. L. M. Carvalho, J. V. A. Goncalves, E. D. da Silva, J. Math. Anal. Appl., 426 (2015), 466–483].

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Keywords

• Orlicz-Sobolev spaces
• mountain pass theorem
• symmetric mountain theorem.

MSC

•  35J20
•  35J50
•  35J55

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