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

Abstract We consider the ground state solutions of the Lane–Emden system with Henon-type weights − Δ u = | x | β | v | q − 1 v , − Δ v = | x | α | u | p − 1 u in the unit ball B of R N 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 R N . We also give precise conditions on the parameters α , β , p and q under which the ground state solutions are not radially symmetric.