Radial symmetry of positive solutions involving the fractional Laplacian

The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem (-\Delta)^{\alpha} u=f(u)+g,\ \ {\rm{in}}\ \ B_1, \quad u=0\ \ {\rm in}\ \ B_1^c, where $(-\Delta)^\alpha$ denotes the fractional Laplacian, $\alpha\in(0,1)$, and $B_1$ denotes the open unit ball centered at the origin in $\R^N$ with $N\ge2$. The function $f:[0,\infty)\to\R$ is assumed to be locally Lipschitz continuous and $g: B_1\to\R$ is radially symmetric and decreasing in $|x|$. In the second place we consider radial symmetry of positive solutions for the equation (-\Delta)^{\alpha} u=f(u),\ \ {\rm{in}}\ \ \R^N, with $u$ decaying at infinity and $f$ satisfying some extra hypothesis, but possibly being non-increasing. Our third goal is to consider radial symmetry of positive solutions for system of the form (-\Delta)^{\alpha_1} u=f_1(v)+g_1,\ \ \ \ & {\rm{in}}\quad B_1,\\[2mm] (-\Delta)^{\alpha_2} v=f_2(u)+g_2,\ \ \ \ & {\rm{in}} \quad B_1,\\[2mm] u=v =0,\ \ \ \ & {\rm{in}}\quad B_1^c, where $\alpha_1,\alpha_2\in(0,1)$, the functions $f_1$ and $f_2$ are locally Lipschitz continuous and increasing in $[0,\infty)$, and the functions $g_1$ and $g_2$ are radially symmetric and decreasing. We prove our results through the method of moving planes, using the recently proved ABP estimates for the fractional Laplacian. We use a truncation technique to overcome the difficulty introduced by the non-local character of the differential operator in the application of the moving planes.