Existence of singular solutions for a Dirichlet problem containing a Dirac mass

Abstract

We give general existence results of solutions (u, v) to the Dirichlet problem

{-Delta u = f (x, u, v) + c delta(0), -Delta u = g(x, u, v) + d delta(0) u = v = 0 on partial derivative B, in D'(B), (P)

where B is the unit ball centered at zero in R(N), N >= 3, delta(0) is the Dirac mass at 0 and c, d are nonnegative constants. No assumptions on the sign of the functions f and g are required. We also characterize the set of (c, d) such that problem (P) admits a solution in some particular cases of the nonlinearities f and g. (C) 2011 Elsevier Ltd. All rights reserved.