The Landis conjecture via Liouville comparison principle and criticality theory

We give partial affirmative answers to Landis conjecture in all dimensions for two different types of linear, second order, elliptic operators in a domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$. In particular, we provide a sharp decay criterion that ensures when a solution of a nonnegative Schrödinger equation in ${\mathbb{R}}^N$ with a potential $V\le 1$ 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.