Direct methods for pseudo-relativistic Schr\"{o}dinger operators

In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schr"{o}dinger operators $(-\mathrm{\Delta }+m^2{)}^s$ with $s\in (0,1)$ and mass $m>0$. As a consequence, we also derive multiple applications of these direct methods. For instance, we prove monotonicity, symmetry and uniqueness results for solutions to various equations involving the operators $(-\mathrm{\Delta }+m^2{)}^s$ in bounded domains, epigraph or ${\mathbb{R}}^N$, including pseudo-relativistic Schr"odinger equations, 3D boson star equations and the equations with De Giorgi type nonlinearities.