Direct Methods for Pseudo-relativistic Schrodinger 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 Schrodinger operators (-Delta + m(2))(s) with s is an element of (0, 1) and mass m > 0. As a consequence, we also derivemultiple 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 Schrodinger 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.