Radial Symmetry and Monotonicity of Solutions to a System Involving Fractional p-Laplacian in a Ball

Abstract

In this paper, we study a nonlinear system involving the fractional p-Laplacian in a unit ball and establish the radial symmetry and monotonicity of its positive solutions. By using the direct method of moving planes, we prove the following result. For 00, if u and v satisfy the following nonlinear system -Δpsux=fvx; -Δptvx=gux, x∈B10; ux,vx=0, x∉B10. and f,g are nonnegative continuous functions satisfying the following: (i) f(r) and g(r) are increasing for r>0; (ii) f′(r)/rp-2, g′(r)/rp-2 are bounded near r=0. Then the positive solutions (u,v) must be radially symmetric and monotone decreasing about the origin.