The Moving Planes Method In Integral Forms To Study A Class Of The Nonlinear Partial Differential Equations(System)

The thesis investigates the symmetry and the monotonicity for the solutions to a class of the nonlinear partial differential equations (system) on bounded domain, ex-terior domain and R+n,respectively. Furthermore,we get the Liouville type theorems for the solutions of the equations (system) on the unbounded domain.The method of moving planes was applied for the PDEs. In order to let the method valid we need the maximum principle and the standard boundary point lem-ma which are both local properties of the differential operators. However,there are no corresponding maximum principles for the class of nonlinear partial differential equations (system) which we consider in this paper. In order to avoid using the lo-cal properties, instead of traditional method of moving planes we get results through the method of moving planes in integral forms. At first, we can rewrite these non-linear partial differential equations (system) as the corresponding integral equations(system). Obviously, the characters satisfied by the solutions of integral equations(system) are also establishment for the solutions of the corresponding nonlinear par-tial differential equations (system). For the results we need, we use the method of moving planes in integral forms for the integral equations (system).In Chapter 1, we are concerned with the nonlinear partial differential equation on bounded domain to get the symmetry for the solutions and the domain. At the same time we also get the monotonicity for the solutions. In Chapter 2, we get the monotonicity and symmetry for the solutions of a system of nonlinear partial differ-ential equations on exterior domain. In Chapter 3, we consider the monotonicity and symmetry for the solutions of a system of nonlinear partial differential equations on exterior domain which is more specific than those in Chapter 2. Further, we can obtain the corresponding Liouville type theorem. In Chapter 4, we obtain the Liouville type theorems for the system of fractional Laplacian equations on R+n.