Pointwise estimates of solutions to semilinear elliptic equations and inequalities

We obtain sharp pointwise estimates for positive solutions to the equation $-Lu+V{u}^q=f$, where $L$ is an elliptic operator in divergence form, $q\in \mathbb{R}\setminus \{0\}$, $f\ge 0$ and $V$ is a function that may change sign, in a domain $\mathrm{\Omega }$ in ${\mathbb{R}}^n$, or in a weighted Riemannian manifold.