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 ∈ ( 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 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.