Monotonicity and symmetry of solutions to fractional $p$-laplacian equations

Abstract

We investigate the fractional $p$-Laplacian equation ${\left(-\mathrm{\Delta }\right)}_p^{s}u=f\left(x,u,\nabla u\right)$. We obtain the monotonicity and symmetry of positive solutions of the fractional $p$-Laplacian equation on bounded and unbounded domains. Specially, for unbounded case, we present a new decay at infinity.