Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Abstract

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials $V(x)\sim |x{|}^{-\alpha }$, $0<\alpha <2$, and $K(x)\sim |x{|}^{-\beta }$, $\beta >0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\epsilon }$ belonging to $W^{1,2}({\mathbb{R}}^n)$ is proved under the assumption that $\sigma <p<(N+2)/(N-2)$ for some $\sigma ={\sigma }_{N,\alpha ,\beta }$. Furthermore, it is shown that $v_{\epsilon }$ are spikes concentrating at a minimum of $\mathcal{A}=V^{\theta }{K}^{-2/(p-1)}$, where $\theta =(p+1)/(p-1)-1/2$.