Symmetry and nonexistence of positive solutions to fractional <i>p</i>-Laplacian equations

In this paper, we consider the fractional p-Laplacian equation \begin{document}$( - \Delta )_p^su(x) = f(u(x)), $ \end{document} where the fractional p-Laplacian is of the form \begin{document}$( - \Delta )_p^su(x) = {C_{n, s, p}}PV\int_{{\mathbb{R}^n}} {\frac{{{{\left| {u(x) - u(y)} \right|}^{p - 2}}(u(x) - u(y))}}{{{{\left| {x - y} \right|}^{n + sp}}}}} dy.$ \end{document} By proving a narrow region principle to the equation above and extending the direct method of moving planes used in fractional Laplacian equations, we establish the radial symmetry in the unit ball and nonexistence on the half space for the solutions, respectively.