Fractional p-Laplacian problems with negative powers in a ball or an exterior domain

Abstract

This paper is concerned with the qualitative properties of positive solutions to fractional p-Laplacian problems

$$\begin{aligned}(-\Delta )^s_p u + |x|^{-\alpha }u^{-q} = 0\end{aligned}$$

in $B_1 \setminus \{0\}$ and $\mathbb {R}^n \setminus \overline{B_1}$, where $0<s<1$, $p\ge 2$, $q>0$, $\alpha \ge 0$ and $B_1$ is the unit ball centered at the origin. By deriving a decay at infinity principle and exploiting the direct method of moving planes for the fractional p-Laplacian, we prove the symmetry or monotonicity of positive solutions to the above problems.