Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Source 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_{\varepsilon} 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_{\varepsilon} are spikes concentrating at a minimum of {\mathcal A}=V^{\theta}K^{-2/(p-1)} , where \theta= (p+1)/(p-1)-1/2 .