Generalized Harnack inequality for semilinear elliptic equations

TL;DR

This paper establishes a sharp Harnack inequality for nonnegative solutions of semilinear divergence form elliptic equations, linking it to the Keller-Osserman condition for entire solutions.

Contribution

It introduces a generalized Harnack inequality for semilinear elliptic equations, extending classical results and connecting to the Keller-Osserman condition.

Findings

Proves a sharp Harnack inequality for nonnegative solutions.

Links the inequality to the Keller-Osserman condition.

Provides insights into the existence of entire solutions.

Abstract

This paper is concerned with semilinear equations in divergence form \[ \diver(A(x)Du) = f(u) \] where $f :\R \to [0,\infty)$ is nondecreasing. We prove a sharp Harnack type inequality for nonnegative solutions which is closely connected to the classical Keller-Osserman condition for the existence of entire solutions.