MULTIPLICITY OF SOLUTIONS FOR BIHARMONIC ELLIPTIC SYSTEMS INVOLVING CRITICAL NONLINEARITY

Lu, Dengfeng; Xiao, Jianhai

doi:10.4134/BKMS.2013.50.5.1693

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MULTIPLICITY OF SOLUTIONS FOR BIHARMONIC ELLIPTIC SYSTEMS INVOLVING CRITICAL NONLINEARITY

Journal title : Bulletin of the Korean Mathematical Society
Volume 50, Issue 5, 2013, pp.1693-1710
Publisher : The Korean Mathematical Society
DOI : 10.4134/BKMS.2013.50.5.1693

Title & Authors

MULTIPLICITY OF SOLUTIONS FOR BIHARMONIC ELLIPTIC SYSTEMS INVOLVING CRITICAL NONLINEARITY
Lu, Dengfeng; Xiao, Jianhai;

Abstract

In this paper, we consider the biharmonic elliptic systems of the form ${$$\{{\Delta}^2u=F_u(u,v)+{\lambda}{\mid}u{\mid}^{q-2}u,\;x{\in}{\Omega},\\{\Delta}^2v=F_v(u,v)+{\delta}{\mid}v{\mid}^{q-2}v,\;x{\in}{\Omega},\\u=\frac{{\partial}u}{{\partial}n}=0,\; v=\frac{{\partial}v}{{\partial}n}=0,\;x{\in}{\partial}{\Omega},$$}$ , where ${${\Omega}{\subset}\mathbb{R}^N$}$ is a bounded domain with smooth boundary ${${\partial}{\Omega}$}$ , ${${\Delta}^2$}$ is the biharmonic operator, ${$N{\geq}5$}$ , ${$2{\leq}q$}$ < ${$2^*$}$ , ${$2^*=\frac{2N}{N-4}$}$ denotes the critical Sobolev exponent, ${$F{\in}C^1(\mathbb{R}^2,\mathbb{R}^+)$}$ is homogeneous function of degree ${$2^*$}$ . By using the variational methods and the Ljusternik-Schnirelmann theory, we obtain multiplicity result of nontrivial solutions under certain hypotheses on ${${\lambda}$}$ and ${${\delta}$}$ .

Keywords

biharmonic elliptic system;critical Sobolev exponent;variational method;multiple solutions;

Language

English

Cited by

1time(s) in CrossRef

ON A CLASSIFICATION OF WARPED PRODUCT SPACES WITH GRADIENT RICCI SOLITONS, Korean Journal of Mathematics, 2016, 24, 4, 627

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