Weakly and strongly singular solutions of semilinear fractional elliptic equations

Abstract

Let Ω⊂R^N (N≥2) be a bounded C² domain containing 0, 0<α<1 and 0<p<N/(N−2α). If δ₀ is the Dirac mass at 0 and k>0, we prove that the weakly singular solution u_k of (E_k) (−Δ)^αu+u^p=kδ₀ in Ω, which vanishes in Ω^c, is a classical solution of (E_*) (−Δ)^αu+u^p=0 in Ω\{0} with the same outer data. Let A=[N/2α,1+2α/N) for N=2,3 and (√5−1)/4N<α<1, otherwise, A=∅; we derive that u_k converges to ∞ in whole Ω as k→∞ for p∈(0,1+2α/N)\A, while the limit of u_k is a strongly singular solution of (E_*) for 1+2α/N<p<N/(N−2α).