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

  • Zhenluo Lou
  • Tobias WethEmail author
  • Zhitao Zhang


We consider the Dirichlet problem for the Hénon–Schrödinger system
$$\begin{aligned} -\Delta u + \kappa _1 u = |x|^{\alpha }\partial _u F(u,v), \qquad -\Delta v + \kappa _2 v = |x|^{\alpha }\partial _v F(u,v) \end{aligned}$$
in the unit ball \(\Omega \subset \mathbb {R}^N, N\ge 2\), where \(\alpha \ge 0\) is a parameter and \(F: \mathbb {R}^2 \rightarrow \mathbb {R}\) 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 \(\alpha \rightarrow \infty \), 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 [19] 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.


Symmetry breaking Morse index Hénon–Schrödinger system Ground state solution 

Mathematics Subject Classification

35B06 35J50 35J57 



The authors thank the referees for their careful reading and useful comments and suggestions. Funding was provided by National Natural Science Foundation of China (Grant No. 11331010).


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHenan University of Science TechnologyLuoyangPeople’s Republic of China
  2. 2.Institut für Mathematik, Goethe-Universität FrankfurtFrankfurt a.MGermany
  3. 3.Academy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingPeople’s Republic of China
  4. 4.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China

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