The Study On Existence Of Solutions For Nonlinear Partial Differential Equations With The Fractional Laplacian

Nonlinear partial differential equations are widely used in many application fields.Since the fractional Laplacian has the properties of memory and heredity,the nonlinear partial differential equation with the fractional Laplacian can describe and solve some practical problems more accurately than the integer partial differential equation.The existence of solutions have always been the focus of scholars’ attention.Many scholars have studied the existence and nonexistence of solutions of nonlinear partial differential equations with the fractional Laplacian and have made some achievements.In this thesis,the existence and nonexistence of solutions of two kinds of nonlinear elliptic equations with the fractional Laplacian are studied.On the one hand,the existence and nonexistence of a class of nonlinear elliptic system with the fractional Laplacian are studied in this thesis.Firstly,the nonexistence theorem of the entire solution of this system in subcritical case is proved by using the test function method and the scaling transformation method combined with some basic inequalities.Secondly,the nonexistence theorem of the entire solution in the critical case is proved by using the integral equivalent system and the estimators in the nonexistence theorem proof of the entire solution in the subcritical case.Finally,two important lemmas are proved by Hardy-Littlewood-Sobolev inequality,on the basis of the integral equivalent system and the two important lemmas,the existence and uniqueness of the solutions of the system in the supercritical case are proved by using the contraction mapping principle.Furthermore,the positive,radial and parity properties are proved.On the other hand,we study the existence and nonexistence of solutions for a class of nonlinear elliptic equation with the fractional Laplacian,in which a potential function term and a nonhomogeneous term are added to the right side of the fractional Lane-Emden equation.Firstly,the nonexistence of the entire solution of the equation in the subcritical case is proved by using the test function method.Secondly,by using the integral equivalent equation and the estimator in the nonexistence theorem proof of the entire solution in the subcritical case,the nonexistence of the entire solution in the critical case is proved.Finally,the existence and uniqueness of the entire solution of the equation in the supercritical case is proved by using the contraction mapping principle and the existence conditions of the solution are given.On this basis,some properties of the solution of the equation related to potential function and nonhomogeneous term are proved by using Picard sequence,mathematical induction,Mazur inequality and argue by contradiction.