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

A. Cannone, M. Yu

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}.

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

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 with . 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}.

Paper Structure

This paper contains 6 sections, 16 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and the first part of Theorem \ref{['sec:introduction-main-thm-last']}
  3. The second part of Theorem \ref{['sec:introduction-main-thm-last']}: the unbounded case.
  4. the proof of Theorem \ref{['sec:introduction-main-thm-last']}: the existence of maximizers in the subcritical case
  5. An upper bound for Schwarz symmetric concentrating sequences
  6. Sharpness of the upper limit and convergence from above

Key Result

Theorem 1.1

CC Suppose that $F$ satisfies $(F_0)$ and is increasing on $[0,+\infty)$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 2.1
  • proof
  • ...and 21 more