Critical and subcritical Trudinger-Moser inequalities on complete noncompact Riemannian manifolds

Abstract

Though the sharp Trudinger-Moser inequalities on compact Riemannian manifolds have been known for some times, optimal constants for such inequalities on complete noncompact Riemannian manifolds are still unknown except in some special cases (such as hyperbolic spaces or Hadamard manifolds). In this paper, we will establish the sharp critical and subcritical Trudinger-Moser type inequalities with best constants on any complete and noncompact Riemannian manifold (M,g) whose Ricci curvature and its injectivity radius have lower bounds. (See Theorems 1.3 and 1.4.)More precisely, we will establish the following sharp critical Trudinger-Moser inequality on such manifolds (M,g,dVg):supu∈W1,n(M),||u||1,τ≤1⁡∫Mϕ(αn|u|nn−1)dVg≤C(n,τ), where τ>0, ϕ(t)=∑k=n−1∞tkk!, αn=nωn−11n−1, and ωn−1 is the area of the unit sphere in Rn, ||u||1,τ=(∫Mτ|u|n+|∇u|ndVg)1n. Moreover, we also prove that the following sharp subcritical Trudinger-Moser inequalitysupu∈W1,n(M),||∇u||n≤1⁡1||u||Ln(M)n∫Mϕ(α|u|nn−1)dVg≤C(n,α) in the sense that the supremum is finite for all α<αn, but infinite for α≥αn.The proofs of both the critical and subcritical Trudinger-Moser inequalities rely crucially on the Trudinger-Moser inequality on any compact manifold M with or without boundary, which is of its own interest. The important property we establish in this paper is that the upper bound C(n,M) for the Trudinger-Moser supremum is continuous and monotone increasing with respect to the volume of M. (See Theorems 1.1 and 1.2.) This upper bound property allows us to carry out a symmetrization-free argument from sharp local inequalities on compact manifolds to the optimal global inequalities on complete and noncompact Riemannian manifolds. Our optimal results sharpen substantially the existing ones on such Riemannian manifolds in the literature.