On the embedding of weighted Sobolev spaces with applications to a planar nonlinear Schrödinger equation

In this paper we study the embedding's properties for the weighted Sobolev space HV1(RN) into the Lebesgue weighted space LWτ(RN). Here V and W are diverging weight functions. The different behaviour of V with respect to W at infinity plays a crucial role. Particular attention is paid to the case V=W. This situation is very delicate since it depends strongly on the dimension and, in particular, N=2 is somewhat a limit case. As an application, an existence result for a planar nonlinear Schrödinger equation in presence of coercive potentials is provided.