Existence and multiplicity of solutions for a class of elliptic boundary value problems

Abstract

In this paper, we investigate the existence and multiplicity of solutions for the following elliptic boundary value problems{−Δu+a(x)u=g(x,u)in Ω,u=0on ∂Ω, where g(x,u)=−Ku(x,u)+Wu(x,u). By using the symmetric mountain pass theorem, we obtain two results about infinitely many solutions when g(x,u) is odd in u, K satisfies the pinching condition and W has a super-quadratic growth. Moreover, when the condition “g(x,u) is odd” is not assumed, by using the mountain pass theorem, we also obtain two existence results of one nontrivial weak solution. One of these results generalizes a recent result in Mao, Zhu and Luan (2012) [10].