  In this paper, we consider the nonlinear equation involving the fractional
p-Laplacian with sign-changing potential. This model draws inspiration from De
Giorgi Conjecture. There are two main results in this paper. Firstly, we obtain
that the solution is radially symmetric within the bounded domain, by applying
the moving plane method. Secondly, by exploiting the idea of the sliding
method, we construct the appropriate auxiliary functions to prove that the
solution is monotone increasing in some direction in the unbounded domain. The
different properties of the solution in bounded and unbounded domains are
mainly attributed to the inherent non-locality of the fractional p-Laplacian.
