The Landis conjecture via Liouville comparison principle and criticality theory

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.