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

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2 Citations (Scopus)

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≥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].

Original languageEnglish
JournalJournal of Functional Analysis
DOIs
Publication statusPublished - Jan 1 2019

Keywords

  • Isometry-invariant solutions
  • Isoperimetric inequality
  • Moser–Trudinger inequality
  • Non-compact Riemannian manifold

ASJC Scopus subject areas

  • Analysis

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