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Existence and bounds of positive solutions for a nonlinear Schrödinger system


Authors: Benedetta Noris and Miguel Ramos
Journal: Proc. Amer. Math. Soc. 138 (2010), 1681-1692
MSC (2010): Primary 35J57, 35J50, 58E05
DOI: https://doi.org/10.1090/S0002-9939-10-10231-7
Published electronically: January 12, 2010
MathSciNet review: 2587453
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Abstract: We prove that, for any $ \lambda\in\mathbb{R}$, the system $ -\Delta u +\lambda u = u^3-\beta uv^2$, $ -\Delta v+\lambda v =v^3-\beta vu^2$, $ u,v\in H^1_0(\Omega),$ where $ \Omega$ is a bounded smooth domain of $ \mathbb{R}^3$, admits a bounded family of positive solutions $ (u_{\beta}, v_{\beta})$ as $ \beta \to +\infty$. An upper bound on the number of nodal sets of the weak limits of $ u_{\beta}-v_{\beta}$ is also provided. Moreover, for any sufficiently large fixed value of $ \beta >0$ the system admits infinitely many positive solutions.


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Additional Information

Benedetta Noris
Affiliation: University of Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy
Email: b.noris@campus.unimib.it

Miguel Ramos
Affiliation: Faculty of Science, Centro de Matemática e Aplicações Fundamentais, University of Lisbon, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
Email: mramos@ptmat.fc.ul.pt

DOI: https://doi.org/10.1090/S0002-9939-10-10231-7
Keywords: Elliptic systems, phase segregation, Morse index
Received by editor(s): July 29, 2009
Published electronically: January 12, 2010
Additional Notes: The first author was partially supported by MIUR, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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