Best constants in bipolar L^p-Hardy-type Inequalities

TL;DR

This paper establishes sharp $L^p$ Hardy inequalities with bipolar potentials for $p eq 2$, extending previous results and demonstrating the existence of minimizers in the energy space.

Contribution

It provides the first sharp $L^p$ multipolar Hardy inequalities for bipolar potentials with $p eq 2$, including existence of minimizers and new features at the critical level.

Findings

Sharp $L^p$ Hardy inequalities proved for bipolar potentials.

Existence of minimizers in the energy space established.

New phenomena observed for $p>2$ at the criticality of the p-Laplacian.

Abstract

In this work we prove sharp $L^p$ versions of multipolar Hardy inequalities in the case of a bipolar potential and $p\geq 2$, which were first developed in the case $p=2$ by Cazacu (CCM 2016) and Cazacu&Zuazua (Studies in phase space analysis with applications to PDEs, 2013). Our results are sharp and minimizers do exist in the energy space. New features appear when $p>2$ compared to the linear case $p=2$ at the level of criticality of the p-Laplacian $-\Delta_p$ perturbed by a singular Hardy bipolar potential.