Symmetry and symmetry breaking for ground state solutions of some strongly coupled elliptic systems (Q1932399)

The authors consider the ground state solutions of the Lane-Emden system with Henon type weights \(-\Delta u =|x|^\beta |v|^{q-1}v\), \(-\Delta v =|x|^\alpha |u|^{p-1}u\) in the unit ball of \(\mathbb{R}^N\) with Dirichlet boundary conditions. They show that ground state solutions always have definite sign in the ball and exhibit a foliated Schwarz symmetry with respect to a unit vector of \(\mathbb{R}^N\). They also give precise conditions on the parameters under which the ground state solutions are not radially symmetric.