Rigidity For Nonlinear Equations On Compact Riemannian Manifolds And Application

In this paper,we obtain rigidity results for weighted Laplace,p-Laplace and weighted p-Laplace type equations on compact Riemannian manifolds with the curvature dimension conditions by using of the ellipse and nonlinear flow methods,respectively,where rigidity means that the PDE has only constant solution when a parameter is in a certain range.The main results in this thesis consist of four chapters as follows:Chapter 1,first of all we introduce research background of this paper.Secondly,we list the main research results.Chapter 2,we consider the weighted Laplace type equation-?fv+?/p-2(v-vp-1)=0.We prove the rigidity of the weighted Laplace type equation on compact Riemannian mani-folds with Bakry-Emery Ricci curvature.Chapter 3,we consider p-Laplace type equation-?pv+?p-?/2-p+?(q-p)(p-2+?(p-1)/?-vq-1)=0.First,we gives some necessary Lemmas.Then obtain rigidity results for p-Laplace type equation by using of the ellipse and nonlinear flow methods.Chapter 4,we consider the weighted p-Laplace-Lp,fv+?/?(v-vq-1)=0.We prove the rigidity of the weighted p-Laplace type equation on compact Riemannian manifolds with Bakry-Emery Ricci curvature.