Maximum principles and qualitative properties of solutions for nonlocal double phase operator

In this paper, we are concerned with the following nonlocal double phase problems with a gradient term: Lu(x)=f(x,u,∇u), where L is a nonlocal double phase operator. We first establish various maximum principles for nonlocal double phase operators in bounded or unbounded domains. Together these maximum principles with the direct method of moving planes and direct sliding methods, we further derive qualitative properties of solutions such as Liouville type theorem, monotonicity, symmetry and uniqueness results for solutions to the nonlocal double phase problems in bounded domains, unbounded domains, epigraph, and Rⁿ respectively. We believe that the new ideas and methods employed here can be conveniently applied to study a variety of nonlinear elliptic problems involving other nonlocal operators.