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On a sharp inequality of Adimurthi-Druet type and extremal functions

José Francisco de Oliveira, João Marcos do Ó

Abstract

Our main purpose in this paper is to establish the existence and nonexistence of extremal functions for sharp inequality of Adimurthi-Druet type for fractional dimensions on the entire space. Precisely, we extend the sharp Trudinger-Moser type inequality in (Calc.Var.Partial Differential Equations, \textbf{52} (2015) 125-163) for the entire space. In addition, we perform the two-step strategy of Carleson-Chang together blow up analysis method to ensure the existence of maximizers for the associated extremal problems for both subcritical and critical regimes. We also present a nonexistence result under subcritical regime for some special cases.

On a sharp inequality of Adimurthi-Druet type and extremal functions

Abstract

Our main purpose in this paper is to establish the existence and nonexistence of extremal functions for sharp inequality of Adimurthi-Druet type for fractional dimensions on the entire space. Precisely, we extend the sharp Trudinger-Moser type inequality in (Calc.Var.Partial Differential Equations, \textbf{52} (2015) 125-163) for the entire space. In addition, we perform the two-step strategy of Carleson-Chang together blow up analysis method to ensure the existence of maximizers for the associated extremal problems for both subcritical and critical regimes. We also present a nonexistence result under subcritical regime for some special cases.
Paper Structure (12 sections, 20 theorems, 293 equations)

This paper contains 12 sections, 20 theorems, 293 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminary results
  3. Sharp Trudinger-Moser inequality of Adimurthi-Druet type
  4. Extremal function for Adimurthi-Druet type inequality
  5. Proof of Theorem \ref{['thm-extremal']}: Maximizers for the subcritical case
  6. Proof of Theorem \ref{['thm-cmax']}: Maximizers for the critical case
  7. Step 1:
  8. Step 2:
  9. Case 1:
  10. Case 2:
  11. Case 3:
  12. Proof of Theorem \ref{['thm non-existence']}: Non-existence of maximizers

Key Result

Theorem 1.1

Let $p\ge 2$ and $\alpha=p-1$ and $\theta\ge 0$. Then,

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2: Subcritical case
  • Theorem 1.3: Critical case
  • Theorem 1.4
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 4.1
  • proof
  • ...and 29 more