Symmetry results for decay solutions of semilinear elliptic systems on half spaces

In this paper, we prove that the decaying positive solutions of semi-linear elliptic systems on the half space with Neumann type condition is symmetric with respect to a line orthogonal to the boundary hyperplane. The monotonicity result is also obtained. Our methods employ the Alexandrov–Serrin method of moving planes combined with Hopf’s lemma at a corner. Our result can be applied to the Lane–Emden system and the stationary Schrödinger system.