New geometric aspects of Moser-Trudinger inequalities on Riemannian manifolds: the non-compact case

TL;DR

This paper explores geometric conditions for Moser-Trudinger inequalities on non-compact Riemannian manifolds, establishing new characterizations and proving existence of invariant solutions for related elliptic problems.

Contribution

It provides a new geometric characterization of Moser-Trudinger inequalities on non-compact manifolds and proves existence results for elliptic equations with critical nonlinearities.

Findings

Characterization of Moser-Trudinger inequalities via isoperimetric and curvature conditions

Existence of non-zero invariant solutions on Hadamard manifolds

Sharp consequences for manifolds with non-negative and non-positive curvature

Abstract

In the first part of the paper we investigate some geometric features of Moser-Trudinger inequalities on complete non-compact Riemannian manifolds. By exploring rearrangement arguments, isoperimetric estimates, and gluing local uniform estimates via Gromov's covering lemma, we provide a Coulhon, Saloff-Coste and Varopoulos type characterization concerning the validity of Moser-Trudinger inequalities on complete non-compact $n-$dimensional Riemannian manifolds $(n\geq 2)$ with Ricci curvature bounded from below. Some sharp consequences are also presented both for non-negatively and non-positively curved Riemannian manifolds, respectively. In the second part, by combining variational arguments and a Lions type symmetrization-compactness principle, we guarantee the existence of a non-zero isometry-invariant solution for an elliptic problem involving the $n-$Laplace-Beltrami operator and a critical nonlinearity on $n-$dimensional homogeneous Hadamard manifolds. Our results complement in several directions those of Y. Yang [J. Funct. Anal., 2012].