Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results

Abstract

In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation (-Delta)(alpha)u = u(p) in Omega\{0}, u = 0 in R-N\Omega where p > 1, alpha is an element of (0, 1), Omega is a bounded C-2 domain in R-N containing the origin, N >= 2 alpha and the fractional Laplacian (-Delta)(alpha) is defined in the principle value sense. We prove that any classical solution u of (1) is a very weak solution of (-Delta)(alpha)u = u(p) + k delta(0) in Omega, u = 0 in R-N\Omega for some k >= 0, where delta(0) is the Dirac mass at the origin. In particular, when p >= N/N-2 alpha, we have that k = 0; when p is an element of (1, N/N-2 alpha), u has removable singularity at the origin if k = 0 and if k > 0, u satisfies that lim(alpha -> 0)u(x)vertical bar x vertical bar(N-2 alpha) = c(N,alpha)k, where c(N,alpha) > 0. Furthermore, when p is an element of (1, N<b>/N-2 alpha )5 we show that there exists k* > 0 such that problem (1) has at least two solutions for k is an element of (0, k*), a unique solution for k = k* and no solution for k > k*.