Maximum Principle And Properties Of Positive Solutions For Fractional Differential Equations With Variable Order

In this paper,we study the maximum principle,the narrow-field principle and properties of positive solutions of the fractional Laplacian differential equation of variable order and the fractional p-Laplacian differential equation of variable order in full space,half space and the unit sphere by using the moving plane method and inequality technique.Firstly,we suppose that positive solutions exist for the fractional Laplacian differential equation of variable order and the fractional p-Laplacian differential equation of variable order.Define the appropriate moving plane.The positive solution satisfies certain properties at some boundary point or at infinity,and then moves the hyperplane so that the positive solution preserves.The original properties are maintained until the hyperplane can no longer be moved.The maximum principle and narrow field principle of the fractional Laplacian differential equation of variable order and the fractional p-Laplacian differential equation of variable order are given,and then the properties of positive solutions of the above two kinds of differential equations in full space,half space and the unit sphere are proved by using these principle.The results in this paper generalize the maximum principle of the fractional Laplacian differential equation of variable order and the fractional p-Laplacial differential equation of variable order and the related result of the properties of positive solutions.