Existence and bounds of positive solutions for a nonlinear Schrödinger system
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- by Benedetta Noris and Miguel Ramos PDF
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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.References
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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
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2587453