On semilinear elliptic equations with borderline Hardy potentials

TL;DR

This paper investigates the asymptotic behavior of solutions to semilinear elliptic equations with borderline Hardy potentials, employing variational methods and Almgren-type formulas to characterize solution behavior near singularities.

Contribution

It introduces a suitable variational framework and applies an Almgren-type monotonicity formula to precisely describe solutions' asymptotics in the presence of borderline Hardy potentials.

Findings

Established the asymptotic profile of solutions near singularities.

Developed a variational approach for equations with borderline Hardy potentials.

Applied Almgren-type formulas to analyze solution behavior.

Abstract

In this paper we study the asymptotic behavior of solutions to an elliptic equation near the singularity of an inverse square potential with a coefficient related to the best constant for the Hardy inequality. Due to the presence of a borderline Hardy potential, a proper variational setting has to be introduced in order to provide a weak formulation of the equation. An Almgren-type monotonicity formula is used to determine the exact asymptotic behavior of solutions.