In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation
left(−Δ)αu=upleft in 1emnormalΩ∖false{0false},leftfalse(−normalΔfalse)αu=0left in 1emRN∖normalΩ,where p>1, α∈(0,1), normalΩ is a bounded C2 domain in RN containing the origin, N⩾2α and the fractional Laplacian false(−normalΔfalse)α is defined in the principle value sense. We prove that any classical solution u of is a very weak solution of
left(−Δ)αu=up+kδ0left in 1emnormalΩ,leftfalse(−normalΔfalse)αu=0left in 1emRN∖normalΩfor some k⩾0, where δ0 is the Dirac mass at the origin. In particular, when p⩾NN−2α, we have that k=0; when p∈(1,NN−2α), u has removable singularity at the origin if k=0 and if k>0, u satisfies that
trueprefixlimx→0ufalse(xfalse)|x|N−2α=cN,αk,where cN,α>0.
Furthermore, when p∈(1,NN−2α), we show that there exists k∗>0 such that problem has at least two solutions for k∈(0,k∗), a unique solution for k=k∗ and no solution for k>k∗.