The Maximum Principle Of Two Kinds Of Fractional Differential Equations And Its Application

In this thesis,the maximum principle,the method of moving planes and the theory of differential equation are applied to study the uniqueness and the continuous dependence on the initial boundary conditions of the solutions of Hadamard fractional differential equations with fractional Laplacian operator or uniform elliptic operator,and also used to establish the radial symmetry and nonexistence of the standing waves of the nonlinear fractional Schrodinger equation.The full text is divided into three chapters:In Chapter 1,fractional derivative,fractional Laplacian operator and the main work of this thesis are introduced.In Chapter 2,we study the uniqueness and the continuous dependence on the initial bound-ary conditions of solutions of Hadamard fractional differential equations.First,The extremum principle of Hadamard fractional derivative is established by finite difference method.Secondly,we prove the maximum principle of Hadamard fractional differential equation with fractional Laplacian operator and the multi-index Hadamard fractional differential equation with uniform elliptic operator.Finally,the uniqueness and the continuous dependence on the initial boundary conditions of the solutions are obtained by using the maximum principle.In Chapter 3,we study the properties of the standing waves of the nonlinear fractional Schrodinger equation with Hardy potential and fractional Laplacian operator.Under the con-dition of decay near infinity,we prove the symmetry of the positive solutions of the nonlinear Hardy-Schrodinger equation by using the direct method of moving planes.Under the condition of no decay,by the Kelvin transformation for the positive solutions of the equation and applying the direct method of moving planes,there are found that in the subcritical case,the positive solution of fractional Hardy-Schrodinger equation does not exist;in the critical case,the positive solution is radially symmetric with respect to some point on RN.