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In this paper we prove existence of least energy nodal solutions for theHamiltonian elliptic system with H\'enon-type weights \[ -\Delta u =|x|^{\beta} |v|^{q-1}v, \quad -\Delta v =|x|^{\alpha}|u|^{p-1}u\quad { in }\Omega, \qquad u=v=0 { on } \partial \Omega, \] where $\Omega$ is a boundedsmooth domain in $\mathbb{R}^N$, $N\geq 1$, $\alpha, \beta \geq 0$ and thenonlinearities are superlinear and subcritical, namely \[ 1>\frac{1}{p+1}+\frac{1}{q+1}> \frac{N-2}{N}. \] When $\Omega$ is either a ballor an annulus centred at the origin and $N \geq 2$, we show that thesesolutions display the so-called foliated Schwarz symmetry. It is natural toconjecture that these solutions are not radially symmetric. We provide such asymmetry breaking in a range of parameters where the solutions of the systembehave like the solutions of a single equation. Our results on the above systemare new even in the case of the Lane-Emden system (i.e. without weights). Asfar as we know, this is the first paper that contains results about leastenergy nodal solutions for strongly coupled elliptic systems and their symmetryproperties.

Existence and symmetry of least energy nodal solutions for Hamiltonian elliptic systems