SYMMETRY OF SOLUTIONS TO SEMILINEAR EQUATIONS INVOLVING THE FRACTIONAL LAPLACIAN

Let 0 < alpha < 2 be any real number. Let Omega subset of R-n be a bounded or an unbounded domain which is convex and symmetric in x(1) direction. We investigate the following Dirichlet problem involving the fractional Laplacian: { (-Delta)(alpha/2)u(x) = f(x,u), x epsilon Omega, u(x) 0, x is not an element of Omega, (1) Applying a direct method of moving planes for the fractional Laplacian, with the help of several maximum principles for anti-symmetric functions, we prove the monotonicity and symmetry of positive solutions in x(1) direction as well as nonexistence of positive solutions under various conditions on f and on the solutions u. We also extend the results to some more complicated cases.