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

TL;DR

This paper establishes a precise threshold for the existence of extremal functions in Trudinger-Moser inequalities with logarithmic kernels across dimensions, revealing their connection to coupled N-Laplacian Schrödinger and fractional Poisson equations.

Contribution

It extends the understanding of extremal functions and thresholds in Trudinger-Moser inequalities to all dimensions greater than 2, including new Euler-Lagrange equations.

Findings

Identifies a sharp dimension-dependent threshold for extremal functions.

Derives Euler-Lagrange equations for extremal functions involving coupled N-Laplacian and fractional equations.

Extends previous results to higher dimensions beyond 2.

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.