On eigenvalue problems involving the critical Hardy potential and Sobolev type inequalities with logarithmic weights in two dimensions

TL;DR

This paper investigates the eigenvalue problem for the Laplacian with Hardy potential in 2D, establishing existence and asymptotic behavior of eigenfunctions using novel Sobolev inequalities with logarithmic weights.

Contribution

It introduces a new Sobolev inequality with logarithmic weights and applies it to analyze eigenfunctions for the Hardy potential problem in two dimensions.

Findings

Existence of the second eigenfunction was proved.

Asymptotic behavior of eigenfunctions near the origin was characterized.

A new Sobolev inequality with a logarithmic weight was established.

Abstract

We consider the two-dimensional eigenvalue problem for the Laplacian with the Neumann boundary condition involving the critical Hardy potential. We prove the existence of the second eigenfunction and study its asymptotic behavior around the origin. A key tool is the Sobolev type inequality with a logarithmic weight, which is shown in this paper as an application of the weighted nonlinear potential theory.