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

We give general existence results of solutions (u,v)(u,v) to the Dirichlet problem equation(P){−Δu=f(x,u,v)+cδ0,−Δv=g(x,u,v)+dδ0in D′(B),u=v=0on∂B, where BB is the unit ball centered at zero in RNRN, N≥3N≥3,δ0δ0 is the Dirac mass at 0 and c,dc,d are nonnegative constants. No assumptions on the sign of the functions ff and gg are required. We also characterize the set of (c,d)(c,d) such that problem (P) admits a solution in some particular cases of the nonlinearities ff and gg.