Monotonicity and symmetry of solutions for the fractional g-Laplacian equation

Abstract

In this paper, we focus on the fractional g-Laplacian equation${(-{\Delta }_g)}^{s}u\left(x\right)=f(x,u,\nabla u)$ in a bounded domain which is strictly convex in the $x_1$-direction and symmetric about the plane $x_1=0$. We also investigate the same equation in the whole space $R^n$. We implement the direct method of moving planes to obtain the monotonicity and symmetry of positive solutions in the $x_1$-direction.