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

Research output: Contribution to journalArticle

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

Fingerprint

Riemannian Manifold
n-dimensional
Hadamard Manifolds
Invariant Solutions
Homogeneous Manifold
Noncompact Manifold
Isoperimetric
Laplace-Beltrami Operator
Symmetrization
Gluing
Uniform Estimates
Ricci Curvature
Rearrangement
Isometry
Elliptic Problems
Compactness
Lemma
Covering
Complement
Nonlinearity

Keywords

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

ASJC Scopus subject areas

  • Analysis

Cite this

@article{12c99d46d13e4ebb8c75443313c130de,
title = "New geometric aspects of Moser–Trudinger inequalities on Riemannian manifolds: the non-compact case",
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].",
keywords = "Isometry-invariant solutions, Isoperimetric inequality, Moser–Trudinger inequality, Non-compact Riemannian manifold",
author = "A. Krist{\'a}ly",
year = "2019",
month = "1",
day = "1",
doi = "10.1016/j.jfa.2019.01.008",
language = "English",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - New geometric aspects of Moser–Trudinger inequalities on Riemannian manifolds

T2 - the non-compact case

AU - Kristály, A.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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].

AB - 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].

KW - Isometry-invariant solutions

KW - Isoperimetric inequality

KW - Moser–Trudinger inequality

KW - Non-compact Riemannian manifold

UR - http://www.scopus.com/inward/record.url?scp=85061116401&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061116401&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2019.01.008

DO - 10.1016/j.jfa.2019.01.008

M3 - Article

AN - SCOPUS:85061116401

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

ER -