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Radial and non radial solutions for Hardy-Hénon type elliptic systems. (English) Zbl 1193.35035

Summary: We discuss existence and non-existence of positive solutions for the following system of Hardy and Hénon type:
\[ \begin{cases} -\Delta v=|x|^{\alpha}u^{p},\,-\Delta u=|x|^{\beta}v^{q} &\text{in } \Omega,\\ u=v=0 &\text{on }\partial \Omega, \end{cases} \]
where \({\Omega\ni 0}\) is a bounded domain in \(\mathbb R^N\), \(N\geq 3\), \(p,q>1\), and \(\alpha,\beta> -N\). We also study symmetry breaking for ground states when \(\Omega \) is the unit ball in \(\mathbb R^N\).

MSC:

35J47 Second-order elliptic systems
35J61 Semilinear elliptic equations
35J50 Variational methods for elliptic systems
35B06 Symmetries, invariants, etc. in context of PDEs
35B09 Positive solutions to PDEs
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