Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian

In this paper, we consider the following nonlinear Schrödinger system involving the fractional Laplacian operator: where $$ . When Ω is the unit ball or $$ , we prove that the solutions $$ are radially symmetric and decreasing. When Ω is the parabolic domain on $$ , we prove that the solutions $$ are increasing. Furthermore, if Ω is the $$ , then we also derive the nonexistence of positive solutions to the system on the half-space. We assume that the nonlinear terms f, g and the solutions u, v satisfy some amenable conditions in different cases.