Some Applications Of The Methods Of Moving Planes And Moving Spheres In Elliptic Equations

This paper studies a priori estimates and existence of positive solutions of generalized mean curvature equations using the methods of moving planes and blowing up, the symmetry and monotonicity properties for positive solutions in systems of semilinear elliptic equations on annular regions using the methods of moving spheres. The whole paper is arranged as follow:In chapter 1, we introduce the history of moving planes and moving spheres and some outstanding applications relied on them. We also give a summary of our ideas on studying the generalized mean curvature equations and the systems of elliptic equations.In chapter 2, we collect some basic materials in partial differential equation and some important theorems which will be used in the next chapter such as maximum theorems and fixed point theorems.In chapter 3, we consider a priori estimates and existence theorems for the Dirichlet problem of generalized mean curvature equations on smooth and strict convex domains. We use the methods of moving planes and blowing up to get L~∞estimates of positive solutions. Then combining the interior gradient estimates, global gradient estimates of quasilinear elliptic equations with the fixed point theorems, we prove the existence theorems.In chapter 4, we consider the properties for the solutions of the systems of generalized elliptic equations on annulus regions with the moving spheres methods and get forms of the symmetry and monotonicity of positive solutions.