Boundary Singularities on a Wedge-like Domain of a Semilinear Elliptic Equation

Abstract

Let n >= 2 and let Omega subset of Rn+1 be a Lipschitz wedge-like domain. We construct positive weak solutions of the problem

Delta u + u(P) = 0 in Omega

that vanish in a suitable trace sense on partial derivative Omega, but which are singular at a prescribed 'edge' of Omega if p is equal to or slightly above a certain exponent p(0) > 1 that depends on Omega. Moreover, for the case in which Omega is unbounded, the solutions have fast decay at infinity.