Multiple solutions for the fourth-order elliptic equation with vanishing potential

Abstract

This paper is concerned with the following fourth-order elliptic equation Δ 2 u − Δ u + V ( x ) u = K ( x ) f ( u ) + μ ξ ( x ) | u | p − 2 u , x ∈ R N , u ∈ H 2 ( R N ) , where Δ 2 ≔ Δ ( Δ ) is the biharmonic operator, N ≥ 5 , V , K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. By working in weighted Sobolev spaces and using a variational method, we prove that the above equation has two nontrivial solutions.