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 \((-\Delta +m^{2})^{s}\) 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 \((-\Delta +m^{2})^{s}\) in bounded or unbounded domains with certain geometrical structures (e.g., coercive epigraph and epigraph), including pseudo-relativistic Schrödinger equations, 3D boson star equations and the equations with De Giorgi-type nonlinearities. When \(m=0\) and \(s=1\), equations with De Giorgi-type nonlinearities are related to De Giorgi conjecture connected with minimal surfaces and the scalar Ginzburg–Landau functional associated to harmonic map.

在本文中，我们建立了各种最大原理，并为涉及物理上有趣的（非局部）伪相对论Schrödinger运算符\（（-\ Delta + m ^ {2}）^ {s} \的方程式开发了直接移动平面和滑动方法。 ）与\（s \ in（0,1）\）和质量\（m> 0 \）。结果，我们也衍生了这些直接方法的多种应用。例如，我们证明了涉及算子\（（-\ Delta + m ^ {2}）^ {s} \）的各种方程的解的单调性，对称性和唯一性结果在具有某些几何结构（例如，强制顽固的硬核和硬核的硬核）的有界或无界域中，包括伪相对论Schrödinger方程，3D玻色子星方程和具有De Giorgi型非线性的方程。当\（m = 0 \）和\（s = 1 \）时，具有De Giorgi型非线性的方程与与最小曲面连接的De Giorgi猜想和与谐波映射相关的标量Ginzburg-Landau函数相关。