Radial and non radial solutions for Hardy-Hénon type elliptic systems

We discuss existence and non-existence of positive solutions for the following system of Hardy and Hénon type: $$\left\{\begin{array}{ll} {-\Delta v=|x|^{\alpha}u^{p},\,-\Delta u=|x|^{\beta}v^{q} \,\,{\rm in}\, \Omega,}\\ {u=v=0 \quad\quad\quad\quad\quad\quad\quad\quad\quad{\rm on}\, \partial \Omega}, \end{array}\right.$$ where $${\Omega\ni 0}$$ is a bounded domain in $${\mathbb{R}^{N}}$$ , N ≥ 3, p, q > 1, and α, β > −N. We also study symmetry breaking for ground states when Ω is the unit ball in $${\mathbb{R}^{N}}$$ .