Ground states for the superlinear Schrödinger equation involving parametric potential

Abstract

We consider the existence of ground states for the following Schrödinger equation −Δu+Vλ(x)u=f(x,u),x∈RN, where f satisfies general superlinear (but subcritical) assumptions, Vλ(x)=λV(x)≥0 and the set {V<V0}≔{x∈RN:V(x)<V0} is nonempty and has finite measure for some V0>0. In particular, if V−1(0) has nonempty interior, the parametric potential Vλ(x) is the classical steep potential well. We point out that this point is the key to verifying that the (PS)c sequence satisfies the local (PS)c condition in existing works. In present paper, we removed this condition by establishing a more fine local (PS)c condition. This work partially solves an open problem proposed by Bartsch and Wang (1995).