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

We consider the ground state solutions of the Lane–Emden system with Hénon-type weights −Δu=|x|β|v|q−1v, −Δv=|x|α|u|p−1u in the unit ball B of RN with Dirichlet boundary conditions, where N⩾1, α,β⩾0, p,q>0 and 1/(p+1)+1/(q+1)>(N−2)/N. We show that such ground state solutions u, v always have definite sign in B and exhibit a foliated Schwarz symmetry with respect to a unit vector of RN. We also give precise conditions on the parameters α, β, p and q under which the ground state solutions are not radially symmetric.