Symmetry breaking via Morse index for equations and systems of H\'enon-Schr\"odinger type

We consider the Dirichlet problem for the Schr"odinger-H'enon system $$-\mathrm{\Delta }u+{\mu }_{1}u=\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }{\mathrm{\partial }}_{u}F(u,v),-\mathrm{\Delta }v+{\mu }_{2}v=\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }{\mathrm{\partial }}_{v}F(u,v)$$ in the unit ball $\mathrm{\Omega }\subset {\mathbb{R}}^N,N\ge 2$, where $\alpha >-1$ is a parameter and $F:{\mathbb{R}}^2\to \mathbb{R}$ is a $p$-homogeneous $C^2$-function for some $p>2$ with $F(u,v)>0$ for $(u,v)≠(0,0)$. We show that, as $\alpha \to \infty$, the Morse index of nontrivial radial solutions of this problem (positive or sign-changing) tends to infinity. This result is new even for the corresponding scalar H'enon equation and extends a previous result by Moreira dos Santos and Pacella for the case $N=2$. In particular, the result implies symmetry breaking for ground state solutions, but also for other solutions obtained by an $\alpha$-independent variational minimax principle.