Multiple Solutions for Degenerate Elliptic Systems Near Resonance at Higher Eigenvalues

Abstract

We study the degenerate semilinear elliptic systems of the form $-\text{div}\left(h_1\right(x)\nabla u)=$ $\lambda \left(a\right(x)u+b(x\left)v\right)+F_u(x,u,v),x\in \Omega ,-\text{div}\left(h_2\right(x)\nabla v)=\lambda \left(d\right(x)v+b(x\left)u\right)+F_v(x,u,v),x\in \Omega ,u{|}_{\partial \Omega }=v{|}_{\partial \Omega }=0$, where $\Omega \subset R^N(N\ge 2)$ is an open bounded domain with smooth boundary $\partial \Omega$, the measurable, nonnegative diffusion coefficients $h_1$, $h_2$ are allowed to vanish in $\Omega$ (as well as at the boundary $\partial \Omega$) and/or to blow up in $\overline{\Omega }$. Some multiplicity results of solutions are obtained for the degenerate elliptic systems which are near resonance at higher eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory.