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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 532430, 19 pages
http://dx.doi.org/10.1155/2012/532430
Research Article

Multiple Solutions for Degenerate Elliptic Systems Near Resonance at Higher Eigenvalues

1School of Sciences, Guizhou Minzu University, Guiyang 550025, China
2School of Mathematics and Computer Science, Bijie University, Bijie 551700, China

Received 24 February 2012; Accepted 29 April 2012

Academic Editor: Shaoyong Lai

Copyright © 2012 Yu-Cheng An and Hong-Min Suo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the degenerate semilinear elliptic systems of the form − d i v ( ℎ 1 ( 𝑥 ) ∇ 𝑢 ) = 𝜆 ( 𝑎 ( 𝑥 ) 𝑢 + 𝑏 ( 𝑥 ) 𝑣 ) + 𝐹 𝑢 ( 𝑥 , 𝑢 , 𝑣 ) , 𝑥 ∈ Ω , − d i v ( ℎ 2 ( 𝑥 ) ∇ 𝑣 ) = 𝜆 ( 𝑑 ( 𝑥 ) 𝑣 + 𝑏 ( 𝑥 ) 𝑢 ) + 𝐹 𝑣 ( 𝑥 , 𝑢 , 𝑣 ) , 𝑥 ∈ Ω , 𝑢 | 𝜕 Ω = 𝑣 | 𝜕 Ω = 0 , where Ω ⊂ 𝑅 𝑁 ( 𝑁 ≥ 2 ) is an open bounded domain with smooth boundary 𝜕 Ω , the measurable, nonnegative diffusion coefficients ℎ 1 , ℎ 2 are allowed to vanish in Ω (as well as at the boundary 𝜕 Ω ) and/or to blow up in Ω . 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.