MULTIPLICITY OF SOLUTIONS FOR BIHARMONIC ELLIPTIC SYSTEMS INVOLVING CRITICAL NONLINEARITY

Abstract

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