Radial symmetry of positive solutions involving the fractional Laplacian

Abstract: 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^{\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.