The V\'azquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients

TL;DR

This paper introduces a unified approach to classical elliptic PDE questions, proving maximum principles and decay estimates for equations with unbounded coefficients, extending previous bounded-ingredient results.

Contribution

It develops a new method based on a weak Harnack inequality to handle elliptic PDEs with unbounded lower-order coefficients, simplifying proofs and broadening applicability.

Findings

Established a weak Harnack inequality with optimal dependence on lower-order terms.

Proved maximum principles for elliptic PDEs with unbounded coefficients.

Demonstrated exponential decay of solutions in unbounded domains.

Abstract

We develop a new, unified approach to the following two classical questions on elliptic PDE: the strong maximum principle for equations with non-Lipschitz nonlinearities, and the at most exponential decay of solutions in the whole space or exterior domains. Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish.