Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

TL;DR

This paper studies mixed local and nonlocal Sobolev inequalities with extremals, establishing their uniqueness, connection to singular elliptic problems, and analyzing the existence, regularity, and symmetry of solutions.

Contribution

It introduces new mixed Sobolev inequalities with extremals and links them to singular elliptic problems involving combined local and nonlocal p-Laplace operators.

Findings

Extremals are unique up to a multiplicative constant.

Mixed Sobolev inequalities characterize existence of weak solutions.

Solutions exhibit specific regularity and symmetry properties.

Abstract

In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal $p$-Laplace operator. Moreover, it is proved that the mixed Sobolev inequalities are necessary and sufficient condition for the existence of weak solutions of such singular problems. As a consequence, a relation between the singular $p$-Laplace and mixed local and nonlocal $p$-Laplace equation is established. Finally, we investigate the existence, uniqueness, regularity and symmetry properties of weak solutions for such problems.