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Direct methods for pseudo-relativistic Schrödinger 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 Schrödinger operators $(-Δ+m^{2})^{s}$ 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 $(-Δ+m^{2})^{s}$ in bounded domains, epigraph or $\mathbb{R}^{N}$, including pseudo-relativistic Schrödinger equations, 3D boson star equations and the equations with De Giorgi type nonlinearities.