Boundary singularities for weak solutions of semilinear elliptic problems

Let Ω be a bounded domain in R^N, N ≥ 2, with smooth boundary ∂Ω. We construct positive weak solutions of the problem Δ u + u^p = 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 frac(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 frac(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 frac(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 frac(N, N - 2) for interior singularities.