Multiplicity of solutions for biharmonic elliptic systems involving critical nonlinearity

摘要

In this paper, we consider the biharmonic elliptic systems of the form egin{equation} left{ egin{array}{ll} Delta^{2} u=F_{u}(u,v) +lambda |u|^{q-2}u, quad& xinOmega, Delta^{2} v=F_{v}(u,v) +delta |v|^{q-2}v, quad& xinOmega, u=rac{partial u}{partial n}=0, v=rac{partial v}{partial n}=0,quad& xinpartialOmega, end{array} ight.onumber end{equation} where $Omegasubset {mathbb{R}}^{N}$ is a bounded domain with smooth boundary $partialOmega$, $Delta^{2}$ is the biharmonic operator, $Ngeq 5,2leq q<2^{*}$, $2^{*}=rac{2N}{N-4}$ denotes the critical Sobolev exponent, $Fin C^{1}(mathbb{R}^{2},mathbb{R}^{+})$ 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 $delta$.