Boundary singularities for weak solutions of semilinear elliptic problems

Let Ω be a bounded domain in RN, N 2, with smooth boundary ∂Ω. We construct positive weak solutions of the problem u + up = 0 in Ω, which vanish in a suitable trace sense on ∂Ω, but which are singular at prescribed isolated points if p is equal or slightly above N+1 N−1 . Similar constructions are carried out for solutions which are singular at any given embedded submanifold of ∂Ω of dimension k ∈ [0,N −2], if p equals or it is slightly above N−k+1 N−k−1 , and even on countable families of these objects, dense on a given closed set. The role of the exponent N+1 N−1 (first discovered by Brezis and Turner [H. Brezis, R. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601–614]) for boundary regularity, parallels that of N N−2 for interior singularities. © 2007 Elsevier Inc. All rights reserved