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Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions

  • Angela Pistoia
  • Hugo Tavares
Article

Abstract

In this paper we deal with the nonlinear Schrödinger system
$$\begin{aligned} -\Delta u_i =\mu _i u_i^3 + \beta u_i \sum _{j\ne i} u_j^2 + \lambda _i u_i, \qquad u_1,\ldots , u_m\in H^1_0(\Omega ) \end{aligned}$$
in dimension 4, a problem with critical Sobolev exponent. In the competitive case (\(\beta <0\) fixed or \(\beta \rightarrow -\infty \)) or in the weakly cooperative case (\(\beta \ge 0\) small), we construct, under suitable assumptions on the Robin function associated to the domain \(\Omega \), families of positive solutions which blowup and concentrate at different points as \(\lambda _1,\ldots , \lambda _m\rightarrow 0\). This problem can be seen as a generalization for systems of a Brezis–Nirenberg type problem.

Keywords

Blowup and concentrating solutions Brezis–Nirenberg type problems Competitive and weakly cooperative systems Critical Sobolev Exponent Cubic Schrödinger systems Lyapunov–Schmidt reduction 

Mathematics Subject Classification

35A15 35J20 35J47 

Notes

Acknowledgments

Angela Pistoia was supported by GNAMPA and Sapienza Fondi di Ricerca. Hugo Tavares was partially supported by Fundação para a Ciência e Tecnologia through the program Investigador FCT and the project PEst-OE/EEI/-LA0009/2013, as well as by the ERC Advanced Grant 2013 N.339958 “Complex Patterns for Strongly Interacting Dynamical Systems—COMPAT”.

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Dipartimento di Metodi e Modelli MatematiciUniversità di Roma “La Sapienza”RomaItaly
  2. 2.CAMGSD, Instituto Superior Técnico, Pavilhão de MatemáticaLisboaPortugal

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