  Our goal in this article is to study the global Lorentz estimates for
gradient of weak solutions to $p$-Laplace double obstacle problems involving
the Schr\"odinger term: $-\Delta_p u + \mathbb{V}|u|^{p-2}u$ with bound
constraints $\psi_1 \le u \le \psi_2$ in non-smooth domains. This problem has
its own interest in mathematics, engineering, physics and other branches of
science. Our approach makes a novel connection between the study of
Calder\'on-Zygmund theory for nonlinear Schr\"odinger type equations and
variational inequalities for double obstacle problems.
