Positive harmonically bounded solutions for semi-linear equations

TL;DR

This paper establishes conditions for positive solutions to semi-linear equations involving elliptic, parabolic, or integro-differential operators, within a general probabilistic framework, extending classical boundary value problem results.

Contribution

It provides necessary and sufficient conditions for positive solutions with prescribed boundary behavior in a broad setting of balayage spaces and Hunt processes.

Findings

Characterization of solutions via integral equations

Extension to general operators and processes

Conditions for existence and boundary behavior

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.