Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type

Abstract

We consider the Dirichlet problem for the Schrödinger-Hénon system $$-\Delta u+{\mu }_{1}u=|x{|}^{\alpha }{\partial }_{u}F(u,v),-\Delta v+{\mu }_{2}v=|x{|}^{\alpha }{\partial }_{v}F(u,v)$$ in the unit ball $\Omega \subset {\mathbb{R}}^N,N\ge 2$, 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 $(u,v)\ne (0,0)$. 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 Hénon 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.