Let $L$ be a second order elliptic operator $L$ with smooth coefficients defined on a domain $��$ in $\mathbb{R}^d $, $d\geq3$, such that $L1\leq 0$. We study existence and properties of continuous solutions to the following problem \begin{equation}\label{00} Lu=��(\cdot,u),% & \hbox{in $��$; in the sens of distribution;} \\ \end{equation} in $��,$ where $��$ is a Greenian domain for $L$ {(possibly unbounded)} in $\mathbb{R}^d$ and $��$ is a nonnegative function on $��\times [0,+\infty [$ increasing with respect to the second variable. By means of thinness, we obtain a characterization of $��$ for which \eqref{00} has a nonnegative nontrivial bounded solution.