The existence of ground state solutions for semi-linear degenerate Schrödinger equations with steep potential well

Abstract

In this article, we study the following degenerated Schrödinger equations: { − Δ γ u + λ V ( x ) u = f ( x , u ) in R N , u ∈ E λ , where λ > 0 is a parameter, Δ γ is a degenerate elliptic operator, the potential V ( x ) has a potential well with bottom and the nonlinearity f ( x , u ) is either super-linear or sub-linear at infinity in u . The existence of ground state solution be obtained by using the variational methods.