A Liouville-type theorem for Schrödinger equations with nonnegative potential

Let $u$ be a solution of $\mathrm{\Delta }u=Vu$ on ${\mathbb{R}}^d$, where $V$ be continuous, nonnegative and bounded. We prove that the condition $${\int }_{r_j\le \mathrm{\mid }x\mathrm{\mid }\le r_j+1}\mathrm{\mid }u(x){\mathrm{\mid }}^{2}dx\to 0,$$ along any sequence $(r_j)$, $r_j\nearrow +\mathrm{\infty }$, implies $u\equiv 0$ on ${\mathbb{R}}^d$. In particular, this implies the Landis conjecture for solutions satisfying a sufficiently fast algebraic decay. These results are generalized to exterior domains as well as for a class of nonlinear Schrödinger equations under suitable conditions on the zero set of the potential.