## Abstract and Applied Analysis

### Multiple Solutions for Degenerate Elliptic Systems Near Resonance at Higher Eigenvalues

#### Abstract

We study the degenerate semilinear elliptic systems of the form $-\text{div}({h}_{1}(x)\nabla u)=$ $\lambda (a(x)u+b(x)v)+{F}_{u}(x,u,v),x\in \Omega ,-\text{div}({h}_{2}(x)\nabla v)=\lambda (d(x)v+b(x)u)+{F}_{v}(x,u,v),x\in \Omega ,u{|}_{\partial \Omega }=v{|}_{\partial \Omega }=0$, 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 532430, 19 pages.

Dates
First available in Project Euclid: 4 April 2013

https://projecteuclid.org/euclid.aaa/1365099939

Digital Object Identifier
doi:10.1155/2012/532430

Mathematical Reviews number (MathSciNet)
MR2947670

Zentralblatt MATH identifier
1250.35092

#### Citation

An, Yu-Cheng; Suo, Hong-Min. Multiple Solutions for Degenerate Elliptic Systems Near Resonance at Higher Eigenvalues. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 532430, 19 pages. doi:10.1155/2012/532430. https://projecteuclid.org/euclid.aaa/1365099939

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