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Abstract

Let $L$ be a second order symmetric elliptic operator 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 problem \begin{equation}\label{00} Lu=\varphi(\cdot,u)\tag{$*$} \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 $(*)$ has a nonnegative nontrivial bounded solution.