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The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.

advanced nonlinear studies

Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument