Properties Of Solutions To Several Kinds Of Elliptic Partial Differential Equations Involving The Fractional Laplacian

In the present article, we mainly study the uniqueness, symmetry, monotonicity and nonexistence of solutions for elliptic partial differential equations involving the fractional Laplacian. We mainly investigate 2 kinds of problems, these problems will be expounded respectively in Chapter 2 and 3.In Chapter 1, we introduce the background of the fractional Laplacian, and collect some preliminary knowledge.In section 2, we study the fractional Laplace equation, and derive a basic but very important theorem, i.e. Liouville Theorem for a-harmonic functions.In Chapter 3, we introduce the direct method of moving planes for the fractional Laplacian and some maximum principles for anti-symmetric functions, and systemati-cally investigate symmetry, monotonicity and nonexistence of solutions to semilinear el-liptic partial differential equations.