Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

We deal with a class on nonlinear Schr¨odinger equations (NLS) with potentials V (x) \simeq |x|^{-\alpha} , 0 < \alpha < 2, and K(x) \simeq |x|^{-\beta} , \beta > 0. Working in weighted Sobolev spaces, the existence of ground states belonging to W^{1,2}(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 these are spikes concentrating at a minimum point of A = V^\theta K^{−2/(p−1)}, where \theta = (p + 1)/(p − 1) − 1/2.