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Multiple normalized solutions for a competing system of Schrödinger equations

  • Thomas Bartsch
  • Nicola SoaveEmail author
Article
  • 227 Downloads

Abstract

We prove the existence of infinitely many solutions \(\lambda _1, \lambda _2 \in \mathbb {R}\), \(u,v \in H^1(\mathbb {R}^3)\), for the nonlinear Schrödinger system
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u - \lambda _1 u = \mu u^3+ \beta u v^2 &{} \text {in }\mathbb {R}^3 \\ -\Delta v- \lambda _2 v = \mu v^3 +\beta u^2 v &{} \text {in }\mathbb {R}^3\\ u,v>0 &{} \text {in }\mathbb {R}^3 \\ \int _{\mathbb {R}^3} u^2 = a^2 \quad \text {and} \quad \int _{\mathbb {R}^3} v^2 = a^2, \end{array}\right. } \end{aligned}$$
where \(a,\mu >0\) and \(\beta \le -\mu \) are prescribed. Our solutions satisfy \(u\ne v\) so they do not come from a scalar equation. The proof is based on a new minimax argument, suited to deal with normalization conditions.

Mathematics Subject Classification

35J50 35J15 35J60 

Notes

Acknowledgements

Nicola Soave is partially supported by the project ERC Advanced Grant 2013 No. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”, by the PRIN-2015KB9WPT_010 Grant: “Variational methods, with applications to problems in mathematical physics and geometry”, and by the GNAMPA group.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematisches InstitutJustus-Liebig-Universität GiessenGiessenGermany
  2. 2.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly

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