Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

We deal with a class on nonlinear Schr\"odinger equations \eqref{eq:1} with potentials $V(x)\sim |x|^{-\a}$, $0 0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\e}$ belonging to $W^{1,2}(\Rn)$ is proved under the assumption that $p$ satisfies \eqref{eq:p}. Furthermore, it is shown that $v_{\e}$ are {\em spikes} concentrating at a minimum of ${\cal A}=V^{\theta}K^{-2/(p-1)}$, where $\theta= (p+1)/(p-1)-1/2$.