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

Abstract

In this paper we study the embedding's properties for the weighted Sobolev space $H_V^1\left(R^N\right)$ into the Lebesgue weighted space $L_W^{\tau }\left(R^N\right)$. 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.