2018/02/09 by Huyuan Chen, Alexander Quaas · 3 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Nonlinear Differential Equations Analysis #Advanced Mathematical Modeling in Engineering
paper · doi:10.1112/jlms.12104
In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation ( − Δ ) α u = u p in Ω ∖ 0 , ( − Δ ) α u = 0 in R N ∖ Ω , (1)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 1 is a very weak solution of ( − Δ ) α u = u p + k δ 0 in Ω , ( − Δ ) α u = 0 in R N ∖ Ω (2)for 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 1 has at least two solutions for k ∈ ( 0 , k ∗ ) , a unique solution for k = k ∗ and no solution for k > k ∗ .