The Uniqueness Of Positive Solutions Of Nonlinear Elliptic Equations On S～n

In this paper, we mainly prove the following result by integration by parts, based on the Bochner-Lichnerowicz-Weitzenbock formula. Theorem Let u(x.) be a positive solution ofwhere α > 0,α= 1, and △g denotes the Laplace-Beltranii operator associated to the metric g on Sn. Then u is a constant function.(1.9) comes from the research of the self similar solution of the harmonic mean curvature flow. At the same time, using the similar method, we simplify the proof of the following result of Gidas and Spruck[14].Theorem (Gidas-Spruck) Let u(x) be a uon-negative C2 solution ofThenu(x) = 0.