  We address an open problem posed by H. Brezis, M. Marcus and A.C. Ponce in:
Nonlinear elliptic equations with measures revisited. In: Mathematical Aspects
of Nonlinear Dispersive Equations (J. Bourgain, C. Kenig, S. Klainerman, eds.),
Annals of Mathematics Studies, 163 (2007). We prove that for any bounded Borel
measure $\mu$ on a smooth bounded domain $D\subset\mathbb R^d$ and
asymptotically convex non-decreasing non-negative continuous function $g$ on
$\mathbb R$ the sequence of solutions to the semi-linear equation (P): $-\Delta
u+g(u)=\rho_n\ast\mu$ ($\rho_n$ is a mollifier) that is subject to homogeneous
Dirichlet condition, converges to the function that solves (P) with
$\rho_n\ast\mu$ replaced by the reduced measure $\mu^*$ (metric projection onto
the space of good measures). We also provide a corresponding version of this
result without non-negativity assumption on $g$.
