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

TL;DR

This paper investigates how the Morse index of solutions to a Hénon-Schrödinger system in a unit ball increases with the parameter alpha, leading to symmetry breaking of solutions as alpha becomes large.

Contribution

It establishes that the Morse index of radial solutions tends to infinity as alpha increases, demonstrating symmetry breaking in the Hénon-Schrödinger system, extending previous results to systems and higher dimensions.

Findings

Morse index of solutions tends to infinity as alpha approaches infinity.

Symmetry breaking occurs for ground state and other solutions.

Results extend previous scalar Hénon equation findings to systems and higher dimensions.

Abstract

We consider the Dirichlet problem for the Schr\"odinger-H\'enon system $$ -\Delta u + \mu_1 u = |x|^{\alpha}\partial_u F(u,v),\quad \qquad -\Delta v + \mu_2 v = |x|^{\alpha}\partial_v F(u,v) $$ in the unit ball $\Omega \subset \mathbb{R}^N, N\geq 2$, 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 $(u,v) \not = (0,0)$. 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.