Multiple Solutions for Degenerate Elliptic Systems Near Resonance at Higher Eigenvalues

Abstract

We study the degenerate semilinear elliptic systems of the form-div(h1(x)∇u)=λ(a(x)u+b(x)v)+Fu(x,u,v),x∈Ω,-div(h2(x)∇v)=λ(d(x)v+b(x)u)+Fv(x,u,v),x∈Ω,u|∂Ω=v|∂Ω=0, whereΩ⊂RN(N≥2)is an open bounded domain with smooth boundary∂Ω, the measurable, nonnegative diffusion coefficientsh1,h2are allowed to vanish inΩ(as well as at the boundary∂Ω) and/or to blow up inΩ¯. 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.