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

TL;DR

This work extends classical Keller-Osserman theory to semilinear elliptic equations with divergence-form operators that admit locally unbounded coefficients, by working in the natural uniformly local $L^q$ framework with $q>n$. It establishes a Keller-Osserman-type solvability criterion, proves a Vasquez-based strong maximum principle, and derives a generalized Harnack inequality for positive solutions, all in the presence of unbounded lower-order terms. The approach hinges on refined energy estimates (Caccioppoli inequalities) and a level-set ODE framework for the measures of sub- and superlevel sets, enabling precise control despite unbounded coefficients. The results unify and extend the theory of SMP and Harnack in the unbounded-coefficient regime, with potential implications for qualitative properties of nonlinear elliptic PDEs in unbounded media.

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 Vázquez. More generally, we obtain a Harnack inequality for positive solutions of this equation, which extends a result by V. Julin.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4
• Definition 5
• Lemma 6
• Lemma 7
• proof
• Lemma 8
• Proposition 9