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

TLDR: In this article, the Dirichlet problem admits a solution in some particular cases of the nonlinearities f and g. No assumptions on the sign of the functions f and G are required.

Abstract: We give general existence results of solutions ( u , v ) to the Dirichlet problem (P) { − Δ u = f ( x , u , v ) + c δ 0 , − Δ v = g ( x , u , v ) + d δ 0 in  D ′ ( B ) , u = v = 0 on ∂ B , where B is the unit ball centered at zero in R N , N ≥ 3 , δ 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 .