The Landis conjecture via Liouville comparison principle and criticality theory

TL;DR

This paper advances understanding of Landis conjecture by establishing decay criteria for solutions of elliptic equations using Liouville comparison principles and criticality theory, with implications for both linear and quasilinear problems.

Contribution

It provides partial affirmative results for Landis conjecture across all dimensions for specific elliptic operators, introducing sharp decay criteria and extending to quasilinear cases.

Findings

Established decay criteria ensuring trivial solutions for Schrödinger equations with potential V≤1.

Extended Landis conjecture results to quasilinear elliptic problems.

Applied Liouville comparison principles and criticality theory to derive main results.

Abstract

We give partial affirmative answers to Landis conjecture in all dimensions for two different types of linear, second order, elliptic operators in a domain $\Omega\subset \mathbb{R}^N$. In particular, we provide a sharp decay criterion that ensures when a solution of a nonnegative Schr\"odinger equation in $\mathbb{R}^N$ with a potential $V\leq 1$ is trivial. Moreover, we address the analogue of Landis conjecture for quasilinear problems. Our approach relies on the application of Liouville comparison principles and criticality theory.