Positive harmonically bounded solutions for semi-linear equations

Abstract

For open sets $U$ in some space $X$, we are interested in positive solutions to semi-linear equations $ Lu=\varphi(\cdot,u)\mu$ on $U$. Here $L$ may be an elliptic or parabolic operator of second order (generator of a diffusion process) or an integro-differential operator (generator of a jump process), $\mu$ is a positive measure on $U$ and $\varphi$ is an arbitrary measurable real function on $U\times \mathbb{R}^+$ such that the functions $t\mapsto \varphi(x,t)$, $x\in U$, are continuous, increasing and vanish at $t=0$. More precisely, given a measurable function $h\ge 0$ on $X$ which is $L$-harmonic on $U$, that is, continuous real on $U$ with $Lh=0$ on $U$, we give necessary and sufficient conditions for the existence of positive solutions $u$ such that $u=h$ on $X\setminus U$ and $u$ has the same ``boundary behavior'' as $h$ on $U$ (Problem 1) or, alternatively, $u\le h$ on $U$, but $u\not\equiv 0$ on $U$ (Problem 2). We show that these problems are equivalent to problems of the existence of positive solutions to certain integral equations $u+K\varphi(\cdot,u)=g$ on $U$, $K$ being a potential kernel. We solve them in the general setting of balayage spaces $(X,\mathcal{W})$ which, in probabilistic terms, corresponds to the setting of transient Hunt processes with strong Feller resolvent.