Keller-Osserman and Harnack type results for nonlinear elliptic PDE with unbounded ingredients

TL;DR

This paper extends Keller-Osserman and Harnack type results to nonlinear elliptic PDEs with unbounded coefficients, establishing solvability, maximum principles, and inequalities under broader conditions than previously known.

Contribution

It proves the Keller-Osserman theorem and strong maximum principle for operators with unbounded coefficients, expanding the applicability of these classical results.

Findings

Validated Keller-Osserman theorem for unbounded coefficient operators

Confirmed strong maximum principle under optimal integral conditions

Established a Harnack inequality for positive solutions in this setting

Abstract

We show that the classical Keller-Osserman theorem on the solvability of the equation $\mathcal{L}[u] = f(u)$ is valid when $\mathcal{L}$ is a general operator in divergence form with unbounded coefficients in the natural regime of local integrability. This has been open up to now, earlier results concerned operators with locally bounded ingredients. We also settle an open question from \cite{SS21} about the validity of the strong maximum principle for supersolutions of $\mathcal{L}[u] = f(u)$ under the optimal integral condition of V\'azquez. More generally, we obtain a Harnack inequality for positive solutions of this equation, which extends a result by V. Julin.