Some Properties of Positive Solutions for Nonlinear Systems Involving Pseudo-Relativistic Operators

In this paper, we mainly investigate the radial symmetry and monotonicity of positive solutions for a nonlinear system involving pseudo-relativistic operators and fractional derivatives of order (0,1). First, we prove a more general Narrow Region Principle and a Decay at Infinity Principle, which are essential for nonlocal pseudo-relativistic operators. Then, by using the direct method of moving planes, we prove the radial symmetry and radial monotonicity of positive solutions for the nonlinear system in the bounded domain B1(0) and the whole space, respectively. Finally, we show that the positive solutions of the system are strictly monotonically increasing in a Lipschitz coercive epigraph.