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may 2013 Existence of nontrivial weak solutions for a quasilinear elliptic systems with concave-convex nonlinearities
Honghui Yin, Zuodong Yang
Bull. Belg. Math. Soc. Simon Stevin 20(2): 309-328 (may 2013). DOI: 10.36045/bbms/1369316547

Abstract

In this paper, our main purpose is to establish the existence of nontrivial weak solutions to the following systems: $$\left\{ \begin{array}{ll} -\triangle_p u=\lambda V(x)|u|^{r-2}u+F_u(x,u,v),\;\;\; x\in \Omega,\\ -\triangle_p v=\theta V(x)|v|^{r-2}v+F_v(x,u,v),\;\;\; x\in \Omega,\\ u=v=0,\;\;\; x\in \partial\Omega, \end{array} \right.$$ where $\Omega$ is a bounded domain in ${\bf R}^N$, $\lambda,\theta>0$, $\triangle_s u=\mbox{div}(|\nabla u|^{s-2}\nabla u)$ is the s-Laplacian of u. We obtain the existence results in two cases: (i)$1<r<p<N$; (ii)$1<p<r<p^*$. The existence results of solutions are obtained by variational methods.

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Honghui Yin. Zuodong Yang. "Existence of nontrivial weak solutions for a quasilinear elliptic systems with concave-convex nonlinearities." Bull. Belg. Math. Soc. Simon Stevin 20 (2) 309 - 328, may 2013. https://doi.org/10.36045/bbms/1369316547

Information

Published: may 2013
First available in Project Euclid: 23 May 2013

zbMATH: 1272.35089
MathSciNet: MR3082767
Digital Object Identifier: 10.36045/bbms/1369316547

Subjects:
Primary: 35J50 , 35J62 , 35J92

Keywords: concave-convex nonlinearities , Critical exponent , Laplacian operator , Mountain pass theorem

Rights: Copyright © 2013 The Belgian Mathematical Society

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Vol.20 • No. 2 • may 2013
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