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Positive ground states for a system of Schrödinger equations with critically growing nonlinearities

  • Pietro d’Avenia
  • Jarosław Mederski
Article

Abstract

We study the following problem
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda u + u^{2^*-2} v &{}\quad \hbox {in }\quad \Omega ,\\ -\Delta v= \mu v^{2^*-1} + u^{2^*-1} &{}\quad \hbox {in }\quad \Omega ,\\ u> 0,v> 0 &{}\quad \hbox {in }\quad \Omega ,\\ u=v=0 &{}\quad \hbox {on }\quad \partial \Omega , \end{array}\right. } \end{aligned}$$
where \(\Omega \) is a bounded domain of \(\mathbb {R}^N\), \(N\ge 4\), \(2^*=2N/(N-2)\), \(\lambda \in \mathbb {R}\) and \(\mu \ge 0\) and we obtain existence and nonexistence results, depending on the value of the parameters \(\lambda \) and \(\mu \).

Mathematics Subject Classification

35J57 35A01 35B33 35J50 

Notes

Acknowledgments

We would like to thank Giusi Vaira for her valuable comments concerning the problem in case \(N\ge 6\).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Dipartimento di Meccanica, Matematica e ManagementPolitecnico di BariBariItaly
  2. 2.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland

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