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

TL;DR

This paper extends a sharp inequality of Adimurthi-Druet type to fractional dimensions on the entire space, establishing existence and nonexistence results for extremal functions using blow-up analysis.

Contribution

It generalizes the sharp Trudinger-Moser inequality to fractional dimensions and employs a two-step blow-up analysis to determine extremal function existence.

Findings

Existence of extremal functions in critical regimes.

Nonexistence of extremal functions in certain subcritical cases.

Extension of inequalities to fractional dimensions.

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.