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 1 ( − Δ ) α u = u p in Ω ∖ { 0 } , ( − Δ ) α u = 0 in R 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 0 in Ω , ( − Δ ) α u = 0 in R 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 that lim x → 0 u ( x ) | x | N − 2 α = c N , α k ,where c N , α > 0. Furthermore, when p ∈ ( 1 , N N − 2 α ), 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 ∗. [ABSTRACT FROM AUTHOR]