Abstract :
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
