  Let $L$ be a second order elliptic operator $L$ with smooth coefficients
defined on a domain $\Omega $ 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=\varphi(\cdot,u),% & \hbox{in $\Omega $; in the
sens of distribution;} \\ \end{equation} in $\Omega,$ where $\Omega$ is a
Greenian domain for $L$ {(possibly unbounded)} in $\mathbb{R}^d$ and $\varphi $
is a nonnegative function on $\Omega\times [0,+\infty [$ increasing with
respect to the second variable. By means of thinness, we obtain a
characterization of $\varphi$ for which \eqref{00} has a nonnegative nontrivial
bounded solution.
