Boundary singularities for weak solutions of semilinear elliptic problems

Abstract

Let Ω be a bounded domain in RN, N⩾2, with smooth boundary ∂Ω. We construct positive weak solutions of the problem Δu+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+1N−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+1N−k−1, and even on countable families of these objects, dense on a given closed set. The role of the exponent N+1N−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 NN−2 for interior singularities.