Large versus bounded solutions to sublinear elliptic problems

Let $L$ be a second order elliptic operator with smooth coefficients defined on a domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ (possibly unbounded), $d\ge 3$. We study nonnegative continuous solutions $u$ to the equation $L u(x) - \varphi (x, u(x))=0$on$\Omega $, where $\phi$ is in the Kato class with respect to the first variable and it grows sublinearly with respect to the second variable. Under fairly general assumptions we prove that if there is a bounded non zero solution then there is no large solution.