On the pseudo-relativistic double phase problems

In this paper, we are concerned with the pseudo-relativistic double phase problems. We first establish various maximal principles for nonlocal operators. As applications, combined with the direct method of moving planes, we prove the monotonicity and symmetry of positive solutions to pseudo-relativistic double phase problems with Hartree nonlinearities. Additionally, we also prove the monotonicity of positive bounded solutions to pseudo-relativistic double phase Dirichlet problems on half-spaces ℝ₊^N. We believe that the various new ideas and techniques we used here would be very useful to deal with other nonlocal double phase problems.