  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.
