Weakly and strongly singular solutions of semilinear fractional elliptic equations

Let Omega subset of R-N (N >= 2) be a bounded C-2 domain containing 0, 0 < alpha < 1 and 0 < p < N/N-2 alpha. If delta(0) is the Dirac mass at 0 and k > 0, we prove that the weakly singular solution u(k) of (E-k) (-Delta)(alpha)u + u(p) - k delta(0) in Omega, which vanishes in Omega(c), is a classical solution of (E-*) (-Delta)(alpha)u + u(p) = 0 in Omega\{0} with the same outer data. Let A = [N/2 alpha, 1 + 2 alpha/N) for N = 2, 3 and root 5-1/4N < alpha < 1, otherwise, A = phi; we derive that u(k) converges to infinity in whole Omega as k -> infinity for p is an element of (0, 1 + 2 alpha/N)\A, while the limit of u(k) is a strongly singular solution of (E-*) for 1 + 2 alpha/N < p < N/N-2 alpha.