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

In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation 1( − Δ ) α u = u pinΩ ∖ { 0 } ,( − Δ ) α u = 0inR N ∖ Ω ,where p > 1 , α ∈ ( 0 , 1 ) , Ω is a bounded C 2 domain in R N containing the origin, N ⩾ 2 α and the fractional Laplacian( − Δ ) α is defined in the principle value sense. We prove that any classical solution u of is a very weak solution of 2( − Δ ) α u = u p + k δ 0inΩ ,( − Δ ) α u = 0inR N ∖ Ωfor some k ⩾ 0 , where δ 0 is the Dirac mass at the origin. In particular, when p ⩾ N N − 2 α, we have that k = 0 ; when p ∈ ( 1 , N N − 2 α ) , u has removable singularity at the origin if k = 0 and if k > 0 , u satisfies thatlim x → 0 u ( x )| x | N − 2 α = c N , α k , wherec N , α > 0 . Furthermore, when p ∈ ( 1 , N N − 2 α ) , we show that there existsk ∗ > 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 ∗ .