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We discuss existence and non-existence of positive solutions for the following system of Hardy and Hénon type:where 0 is a bounded domain in R N , N ≥ 3, p, q > 1, and α, β > −N . We also study symmetry breaking for ground states when is the unit ball in R N .

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We consider the pendulum type equationwhere the functions Wand h satisfy (HI) WE C 2 (JR. x JR.; JR.) is I-periodic in t and T-periodic in u, (H2) hE C(JR.;JR.) is I-periodic and fol h(t)dt = 0, (H3) Vet, u) = Wet, u) -h(t)u is even in t.