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Equivalence of critical and subcritical sharp Trudinger-Moser inequalities in fractional dimensions and extremal functions

José Francisco de Oliveira, João Marcos do Ó

Abstract

We establish critical and subcritical sharp Trudinger-Moser inequalities for fractional dimensions on the whole space. Moreover, we obtain asymptotic lower and upper bounds for the fractional subcritical Trudinger-Moser supremum from which we can prove the equivalence between critical and subcritical inequalities. Using this equivalence, we prove the existence of maximizers for both the subcritical and critical associated extremal problems. As a by-product of this development, we can explicitly calculate the value of the critical supremum in some special situations.

Equivalence of critical and subcritical sharp Trudinger-Moser inequalities in fractional dimensions and extremal functions

Abstract

We establish critical and subcritical sharp Trudinger-Moser inequalities for fractional dimensions on the whole space. Moreover, we obtain asymptotic lower and upper bounds for the fractional subcritical Trudinger-Moser supremum from which we can prove the equivalence between critical and subcritical inequalities. Using this equivalence, we prove the existence of maximizers for both the subcritical and critical associated extremal problems. As a by-product of this development, we can explicitly calculate the value of the critical supremum in some special situations.

Paper Structure

This paper contains 12 sections, 16 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Sharp subcritical Trudinger-Moser inequality: Proof of Theorem \ref{['thm-AT']}
  3. Some elementary properties
  4. Proof of Theorem \ref{['thm-AT']}
  5. Equivalence of critical and subcritical Trudinger-Moser inequalities
  6. Proof of Theorem \ref{['thm-MR']}
  7. Existence of extremal fuctions
  8. Maximizers for the subcritical Trudinger-Moser inequality
  9. Maximizers for the critical Trudinger-Moser inequality
  10. Case 1:
  11. Case 2:
  12. Proof of Theorem \ref{['thm-MR-Maximizers-analyzed']}

Key Result

Theorem A

Assume $0<R<\infty$, $\alpha\ge 1$, $\theta\ge 0$ and $p=\alpha+1$ be real numbers. Then,

Theorems & Definitions (26)

  • Theorem A
  • Theorem B
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 16 more