On fractional critical problems with multi-polar Hardy potentials

TL;DR

This paper studies positive solutions to fractional equations with multi-polar Hardy potentials and Sobolev critical nonlinearities, overcoming nonlocal and criticality challenges via extended formulations and concentration-compactness.

Contribution

It introduces a novel approach combining extended formulations and asymptotic estimates to analyze existence conditions for solutions in multi-polar Hardy potential problems.

Findings

Existence of solutions depends on masses and distances between poles.

Sharp asymptotic estimates are established for single-pole solutions.

A concentration-compactness argument determines minimizer existence.

Abstract

We investigate the existence of positive solutions to fractional equations presenting a double criticality: a multi-polar Hardy-type potential and a Sobolev critical nonlinearity. The nonlocal nature of the operator and the absence of explicit ground states for the single-pole equation stand as major difficulties. We overcome these obstacles by passing to an extended formulation of the problem and by establishing sharp asymptotic estimates for the solutions in the case of a single pole. Then, through a concentration-compactness argument, we show that the existence of minimizers is dictated by the magnitude of the masses and the mutual distances between the corresponding poles.