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Article type: Research Article
Authors: Chen, Huyuan; | Véron, Laurent;
Affiliations: Department of Mathematics, Jiangxi Normal University, Nanchang, China | Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile. E-mail: chenhuyuan@yeah.net | Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais, Tours, France. E-mail: Laurent.Veron@lmpt.univ-tours.fr
Note: [] Corresponding author. E-mail: Laurent.Veron@lmpt.univ-tours.fr
Abstract: Let Ω⊂RN (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 uk of (Ek) (−Δ)αu+up=kδ0 in Ω, which vanishes in Ωc, is a classical solution of (E*) (−Δ)αu+up=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 uk converges to ∞ in whole Ω as k→∞ for p∈(0,1+2α/N)\A, while the limit of uk is a strongly singular solution of (E*) for 1+2α/N<p<N/(N−2α).
Keywords: fractional Laplacian, Dirac mass, isolated singularity, weak solution, weakly singular solution, strongly singular solution
DOI: 10.3233/ASY-141216
Journal: Asymptotic Analysis, vol. 88, no. 3, pp. 165-184, 2014
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