SOBOLEV INEQUALITIES ON A WEIGHTED RIEMANNIAN MANIFOLD OF POSITIVE BAKRY-EMERY CURVATURE AND CONVEX BOUNDARY

摘要

In this paper, we study some nonlinear elliptic equations on a compact n-dimensional weighted Riemannian manifold of positive m-Bakry-Emery-Ricci curvature and convex boundary. Our main purpose is to find conditions which imply that such elliptic equations admit only constant solutions. As an application, we obtain weighted Sobolev inequalities with explicit constants that extend the inequalities obtained by Ilias [1983; 1996] in the Riemannian setting. In a last part of the article, as applications we derive a new Onofri inequality, a logarithmic Sobolev inequality and estimates for the eigenvalues of a weighted Laplacian and for the trace of the weighted heat kernel.