The maximum principles for fractional Laplacian equations and their applications

This paper is devoted to investigate the symmetry and monotonicity properties for positive solutions of fractional Laplacian equations. Especially, we consider the following fractional Laplacian equation with homogeneous Dirichlet condition: { (-Delta)(alpha/2) u = f(x, u,del u) in Omega, for alpha is an element of(0, 2). u > 0, in Omega; u equivalent to 0, in R-n\Omega, Here Omega is a domain (bounded or unbounded) in R-n which is convex in x(1)-direction. (-Delta)(alpha/2) is the nonlocal fractional Laplacian operator which is defined as (-Delta)(alpha/2)u(x) = Cn,alpha P.V. integral(n)(R) u(x) - u(y)/vertical bar x - y vertical bar (n+alpha), 0 < alpha < 2. Under various conditions on f(x, u, p) and on a solution u(x) it is shown that u is strictly increasing in x(1) in the left half of Omega, or in Omega. Symmetry (in x(1)) of some solutions is proved.