Direct Methods for Pseudo-relativistic Schrödinger Operators

Abstract

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 $$s\in (0,1)$$ 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 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.