Blow-up analysis and extremal functions for nonlocal interaction functionals in dimension $N$

TL;DR

This paper investigates extremal functions for nonlocal energy functionals with logarithmic convolution potentials in higher dimensions, establishing existence results through blow-up analysis in the critical growth case.

Contribution

It extends previous results to higher dimensions and refines existing inequalities for nonlocal functionals with critical growth.

Findings

Proved existence of extremal functions in the critical case.

Extended inequalities to higher dimensions.

Sharpened previous results in the literature.

Abstract

In this paper we study Moser-Trudinger type inequalities for some nonlocal energy functionals in presence of a logarithmic convolution potential, when the domain is a ball of $\mathbb{R}^N$ with $N \geq 2$. In particular, we perform a blow-up analysis to prove existence of extremal functions in the borderline case of critical growth. Using this, we extend the results in \cite{CiWeYu} to higher dimension and sharpen \cite{CC}.