Existence and non existence of the ground state solution for the nonlinear Schroedinger equations with $V(\infty)=0$

Abstract

We study the existence of the ground state solution of the problem $$ \begin{cases} -\Delta u+V(x)u=f'(u) & x\in\mathbb R^N, \\ u(x)> 0, \end{cases} $$ under the assumption that $\lim_{x\to\infty}V(x)=0$.