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

In this paper, we investigate the existence and multiplicity of solutions for the following elliptic boundary value problems { − Δ u + a ( x ) u = g ( x , u ) in Ω , u = 0 on ∂ Ω , where g ( x , u ) = − K u ( x , u ) + W u ( 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].