Qualitative properties of singular solutions to fractional elliptic equations

In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations (-Delta)(alpha)u = f(u), x is an element of Omega\Gamma, where 0 < alpha < 1, Omega = R-N or Omega is a smooth bounded domain, Gamma is a singular subset of n with fractional capacity zero, f (t) is locally bounded and positive for t is an element of [0, infinity), and f (t)/t((N+2 alpha)/(N-2 alpha)) is nonincreasing in t for large t, rather than for every t > 0. Our main result is that the solutions satisfy the estimate f(u(x))/u(x) <= Cd(x, Gamma)(-2 alpha). This estimate is new even for Gamma = {0}. As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.