Generalized solutions to semilinear elliptic equations with measure data

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): $-\mathrm{\Delta }u+g(u)={\rho }_n\ast \mu$ (${\rho }_n$ 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 ${\mu }^{\ast }$ (metric projection onto the space of good measures). We also provide a corresponding version of this result without non-negativity assumption on $g$.