Symmetry of Positive Solutions to Quasilinear Fractional Systems

Abstract

Using the method of moving planes, we establish the radial symmetry of positive solutions to the fractional system \[ \begin{cases} (-\Delta)^s_p u = f(u,v), \\ (-\Delta)^t_q v = g(u,v) \end{cases} \] in the entire Euclidean space $\mathbb{R}^n$ and in the unit ball, where $0 \lt s,t \lt 1$ and $p,q \geq 2$. In particular, our result can be applied to the nonlinearities $f(u,v) \equiv u^a v^b$ and $g(u,v) \equiv u^c v^d$, where $a,d \in \mathbb{R}$ and $b,c \gt 0$.