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

In this paper, we study the fourth order elliptic equations Δ2u−Δu+λV(x)u=f(x,u)Δ2u−Δu+λV(x)u=f(x,u) in RNRN, u∈H2(RN)u∈H2(RN), where λ≥1λ≥1 and the nonlinearity ff 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 RNRN, J. Math. Anal. Appl. 395 (2012) 608–615].