A CONSTRUCTION OF SINGULAR SOLUTIONS FOR A SEMILINEAR ELLIPTIC EQUATION USING ASYMPTOTIC ANALYSIS

The aim of this paper is to prove the existence of weak solutions to t he equation Delta u + u(p) = 0 which are positive in a domain Omega su bset of R(N), vanish at the boundary, and have prescribed isolated sin gularities. The exponent p is required to lie in the interval (N/(N - 2), (N + 2)/(N - 2)). We also prove the existence of solutions to the equation Delta u + u(p) = 0 which are positive in a domain Omega subse t of R(n) and which are singular along arbitrary smooth k-dimensional submanifolds in the interior of these domains provided p lies in the i nterval ((n - k)/(n - k - 2), (n - k + 2)/(n - k - 2)). A particular c ase is when p = (n + 2)/(n - 2), in which case solutions correspond to solutions of the singular Yamabe problem. The method used here is a m ixture of different ingredients used by both authors in their separate constructions of solutions to the singular Yamabe problem, along with a new set of scaling techniques.