Existence and non existence of the ground state solution for the nonlinear Schrödinger equations with \(V(\infty)=0\) (Q2501470)

The authors study the existence of the ground state solution of the problem \[ \Delta u+ V(x)u= f'(u),\quad x\in\mathbb{R}^N,\;u(x)> 0. \] They consider the case where \(\lim_{x\to\infty} V(x)= 0\) under some natural assumptions on \(f\).