MONOTONICITY AND SYMMETRY OF SOLUTIONS TO FRACTIONAL LAPLACIAN EQUATION

Let 0 < alpha < 2 be any real number and let Omega be an open domain in R-n. Consider the following Dirichlet problem of a semi -linear equation involving the fractional Laplacian: {(-Delta)(alpha/2)u(x) = f(x, u, del u), u(x) > 0, x is an element of Omega, u(x) equivalent to 0, x is not an element of Omega. (1) In this paper, instead of using the conventional extension method introduced by Caffarelli and Silvestre, we employ a direct method of moving planes for the fractional Laplacian to obtain the monotonicity and symmetry of the positive solutions of a semi -linear equation involving the fractional Laplacian. By using the integral definition of the fractional Laplacian, we first introduce various maximum principles which play an important role in the process of moving planes. Then we establish the monotonicity and symmetry of positive solutions of the semi -linear equations involving the fractional Laplacian.