Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V(x)~ |x|-α , 0<α<2 , and K(x)~ |x|-β , β>0 . Working in weighted Sobolev spaces, the existence of ground states vε belonging to W1,2( R^n) is proved under the assumption that σ<p<(N+2)/(N-2) for some σ=σN,α,β . Furthermore, it is shown that vε are spikes concentrating at a minimum of { A}=VθK-2/(p-1) , where θ= (p+1)/(p-1)-1/2 .