New geometric aspects of Moser-Trudinger inequalities on Riemannian manifolds: the non-compact case. (English) Zbl 1410.53046
Summary: 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 [ibid. 263, No. 7, 1894–1938 (2012; Zbl 1256.53034)].
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |
Keywords:
Moser-Trudinger inequality; non-compact Riemannian manifold; isoperimetric inequality; isometry-invariant solutionsCitations:
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