Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type
We consider the Dirichlet problem for the Schrödinger-Hénon system $$ -Δu + μ_1 u = |x|^α\partial_u F(u,v),\quad \qquad -Δv + μ_2 v = |x|^α\partial_v F(u,v) $$ in the unit ball $Ω\subset \mathbb{R}^N, N\geq 2$, where $α>-1$ 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 $α\to \infty$, 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 $α$-independent variational minimax principle.