A sharp threshold for Trudinger-Moser type inequalities with logarithmic kernels in dimension N

TL;DR

This work analyzes sharp Trudinger–Moser-type inequalities with logarithmic kernels in dimension $N \ge 2$, on both the unit ball and the whole space. It identifies a sharp, $N$-dependent threshold for the existence of extremals versus blow-up based on the growth of the nonlinearity $G$, and shows that maximizers satisfy Euler–Lagrange equations. In $\mathbb{R}^N$, the limiting Euler–Lagrange problem is a nonlocal logarithmic Choquard-type system that parallels an $N$-Laplacian Schrödinger equation coupled with a higher-order fractional Poisson equation, linking the two formulations. The results extend known planar results to all $N \ge 2$, establish radial symmetry of maximizers via rearrangements, and delineate when extremals exist or blow up, depending on critical-growth thresholds.

Abstract

In the paper we investigate Trudinger-Moser type inequalities in presence of logarithmic kernels in dimension N. A sharp threshold, depending on N, is detected for the existence of estremal functions or blow-up, where the domain is the ball or the entire space. We also show that the extremal functions satisfy suitable Euler-Lagrange equations. When the domain is the entire space, such equation can be derived by N-Laplacian Schrodinger equation strongly coupled with a higher order fractional Poisson's equation. The results extends [16] to any dimension N bigger than 2.

Theorems & Definitions (44)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Corollary 2.2
• Lemma 2.3
• Lemma 2.4
• proof