On the embedding of weighted Sobolev spaces with applications to a planar nonlinear Schr\"{o}dinger equation

TL;DR

This paper investigates the embedding properties of weighted Sobolev spaces into weighted Lebesgue spaces, focusing on the case where the weights are equal, and applies these results to establish existence of solutions for a planar nonlinear Schrödinger equation.

Contribution

It provides new insights into the embedding of weighted Sobolev spaces with equal weights and applies these findings to a specific nonlinear Schrödinger equation in two dimensions.

Findings

Embedding properties depend on the behavior of weights at infinity.

Special case when weights are equal is highly delicate and dimension-dependent.

Existence of solutions for a planar nonlinear Schrödinger equation with coercive potentials is established.

Abstract

In this paper we study the embedding properties for the weighted Sobolev space $H^1_V(\mathbb{R}^N)$ into the Lebesgue weighted space $L^\tau_W(\mathbb{R}^N)$. 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 Schr\"odinger equation in presence of coercive potentials is provided.