Existence and multiplicity of solutions for fourth-order elliptic equations in

In this paper, we study the fourth order elliptic equations Δ 2 u − Δ u + λ V ( x ) u = f ( x , u ) in R N , u ∈ H 2 ( R N ) , where λ ≥ 1 and the nonlinearity f is either superlinear or sublinear at infinity. We establish the existence and multiplicity results without the compactness of embedding of the working space, which unify and sharply improve the recent results of Liu, Chen and Wu [J. Liu, S. Chen, X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in R N , J. Math. Anal. Appl. 395 (2012) 608–615].