A Trudinger-Moser inequality involving L^p -norm on a closed Riemann surface

TL;DR

This paper establishes a new Trudinger-Moser inequality involving L^p-norms on closed Riemann surfaces and proves the existence of extremal functions, extending previous Euclidean space results.

Contribution

It introduces a novel inequality on Riemann surfaces and demonstrates the existence of extremal functions, expanding the scope of earlier Euclidean space findings.

Findings

Established a Trudinger-Moser inequality involving L^p-norm on closed Riemann surfaces.

Proved the existence of extremal functions for the associated functional.

Extended prior results from Euclidean space to Riemann surfaces.

Abstract

In this paper, using the method of blow-up analysis, we obtained a Trudinger-Moser inequality involving L p -norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger-Moser functional. This extends an early result of Yang [27]. Similar result in the Euclidean space was also established by Lu-Yang [17].