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 0<s,t<1,p>0, 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.

Symmetry of Positive Solutions to Quasilinear Fractional Systems

“…Later, Wu and Niu [26] extend the result to any Lipschitz continuous function f (see [26,Theorem 1.2]). A symmetry result for system (1.1) was recently established in [8] under some technical assumptions on f and g. One of our purposes in writing this paper is to improve [8] by establishing the symmetry of positive solutions to (1.1) for a very large class of f and g. Our first result reads as follows.…”

“…Therefore, we go directly to Step 2. Hence, our proof is simpler and requires fewer technical assumptions on f and g comparing to [8,9].…”

Symmetric properties for Choquard equations involving fully nonlinear nonlocal operator

“…In 2015, Chen, Li and Li [11] developed a new technique (the direct method of moving planes) that can be applied for problems with fractional Laplace operator. It is very effective in dealing with equations involving fully nonlinear nonlocal operators or uniformly elliptic nonlocal operators, for example the results in [6,10,33,34,35] and the references therein.…”