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 (-Δ)^α 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 ${\mathbb{R}}^N$ with $N\ge 2$. The function $f:[0,\infty )\to \mathbb{R}$ is assumed to be locally Lipschitz continuous and $g:B_1\to \mathbb{R}$ is radially symmetric and decreasing in $|x|$. In the second place we consider radial symmetry of positive solutions for the equation (-Δ)^α 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 (-Δ)^{α_1} u=f_1(v)+g_1,\ \ \ \ & {\rm{in}}\quad B_1,\\[2mm] (-Δ)^{α_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.

27 pages, Communications in Contemporary Mathematics 2013

References in corpus (3)

• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian
• The Dirichlet problem for the fractional Laplacian: regularity up to the boundary
• Regularity of radial extremal solutions for some non local semilinear equations

Cited by in corpus (2)

• Symmetry via antisymmetric maximum principles in nonlocal problems of variable order
• Symmetry breaking for an elliptic equation involving the Fractional Laplacian