Trudinger-Moser inequalities on complete noncompact Riemannian manifolds

TL;DR

This paper investigates the conditions under which Trudinger-Moser inequalities hold on complete noncompact Riemannian manifolds and applies these results to establish existence of solutions for certain nonlinear equations.

Contribution

It provides the first comprehensive necessary and sufficient conditions for Trudinger-Moser inequalities on noncompact manifolds, extending known results beyond compact cases.

Findings

Established necessary and sufficient conditions for inequalities

Proved existence of solutions for quasilinear equations with exponential nonlinearity

Extended understanding of functional inequalities on noncompact manifolds

Abstract

Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on complete noncompact Riemannian manifolds. For applications of such inequalities, we obtained existence results for some quasilinear equations with nonlinearity of exponential growth.