On a supercritical Hardy-Sobolev type inequality with logarithmic term and related extremal problem

TL;DR

This paper investigates supercritical Hardy-Sobolev inequalities with a logarithmic term, proving the existence of extremal functions and solutions to related elliptic PDEs despite compactness loss.

Contribution

It establishes the existence of extremal functions for a class of supercritical inequalities with logarithmic terms and applies these results to elliptic PDEs.

Findings

Existence of extremal functions proven

Weak solutions to elliptic PDEs established

Addresses compactness issues in supercritical regimes

Abstract

Our main goal is to investigate supercritical Hardy-Sobolev type inequalities with a logarithmic term and their corresponding variational problem. We prove the existence of extremal functions for the associated variational problem, despite the loss of compactness. As an application, we show the existence of weak solution to a general class of related elliptic partial differential equations with a logarithmic term.