QUALITATIVE PROPERTIES OF POSITIVE SOLUTIONS TO FRACTIONAL p-LAPLACIAN EQUATIONS IN AN UNBOUNDED PARABOLIC DOMAIN

In this paper we investigate the qualitative properties of solutions to the nonlinear equations involving fractional p-Laplacian operators (-Delta)(p)(theta)u(x) = f (u(x)) in the unbounded parabolic domain P = {x = (x(1), x') is an element of R-n vertical bar x(1) > vertical bar x'vertical bar(2), x' = (x(2), x(3), ... x(n))}, where (-Delta)(p)(theta), is a nonlocal pseudo-differential operator. First we take advantage of the direct method of moving plane to show the monotonicity of positive solutions to the nonlinear equations in the direction xi. Moreover, we demonstrate the radial symmetry and monotonicity of positive solutions in Rn-1 about the origin, i.e., u(x) = u(x(1), vertical bar x'vertical bar).