Weakly and strongly singular solutions of semilinear fractional elliptic equations

If p ∈ (0, N N−2α ), α ∈ (0, 1), k > 0 and Ω ⊂ R is a bounded C domain containing 0 and δ0 is the Dirac measure at 0, we prove that the weak solution of (E)k (−∆) u + u = kδ0 in Ω which vanishes in Ω is a weak singular solution of (E)∞ (−∆) u + u = 0 in Ω \ {0} with the same outer data. Furthermore, we study the limit of weak solutions of (E)k when k → ∞. For p ∈ (0, 1+ 2α N ], the limit is infinity in Ω. For p ∈ (1 + 2α N , N N−2α ), the limit is a strong singular solution of (E)∞.