Large versus bounded solutions to sublinear elliptic problems

Let $L $ be a second order elliptic operator with smooth coefficients defined on a domain $\varOmega \subset \mathbb {R}^d$ (possibly unbounded), $d\geq 3$. We study nonnegative continuous solutions $u$ to the equation $L u(x) - \varphi (x, u(x))=0$ on $\varOmega $, where $\varphi $ 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 nonzero solution then there is no large solution.