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A Trudinger-Moser inequality under a refined constraint in fractional dimensions and extremal functions

Ruan Diego da Silva Paiva, José Francisco de Oliveira

Abstract

We establish a Trudinger-Moser type inequality with a Tintarev-type constraint in fractional-dimensional spaces and prove the existence of maximizers in the critical regime. Our results provide a refinement of those in (Calc. Var. 52 (2015), 125-163) in the setting of fractional-dimensional spaces, as well as of those in (Ann. Global Anal. Geom. 54 (2018), 237-256) for classical Sobolev spaces.

A Trudinger-Moser inequality under a refined constraint in fractional dimensions and extremal functions

Abstract

We establish a Trudinger-Moser type inequality with a Tintarev-type constraint in fractional-dimensional spaces and prove the existence of maximizers in the critical regime. Our results provide a refinement of those in (Calc. Var. 52 (2015), 125-163) in the setting of fractional-dimensional spaces, as well as of those in (Ann. Global Anal. Geom. 54 (2018), 237-256) for classical Sobolev spaces.

Paper Structure

This paper contains 5 sections, 21 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. The subcritical inequalities and their extremal functions
  3. Boundedness and Extremals in the critical regime
  4. Blow-up analysis
  5. Test-function computations

Key Result

Theorem 1.1

Let $p \geq 2$, $\alpha$ and $R$ satisfy assumption TM-case, and let $\theta\ge \alpha$. Then for any $0\leq \nu<\lambda_{\alpha,\theta}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Lions-type estimate
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 30 more