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Concentration on Curves for a Neumann Ambrosetti–Prodi-Type Problem in Two-Dimensional Domains

  • Sami Baraket
  • Zied  Khemiri
  • Fethi Mahmoudi
  • Abdellatif Messaoudi
Article
  • 13 Downloads

Abstract

Given a smooth bounded domain \(\Omega \subset \mathbb {R}^2 \), we consider the problem
$$\begin{aligned} \left\{ \begin{array} {cccccc} - \Delta u = |u|^p - s\,\psi &{}\hbox {in } \ \Omega \\ \\ \dfrac{\partial u}{\partial \nu } = 0 &{}\hbox {on}\ \partial \Omega \end{array}\right. \end{aligned}$$
where \(p > 1\), \(s > 0\) is a large parameter, \(\psi \) is a positive function and \(\nu \) denotes the outward normal of \(\partial \Omega \). Let \(\Gamma \) be a curve intersecting orthogonally with \(\partial \Omega \) at exactly two points and dividing \(\Omega \) into two parts. Assuming moreover that \(\Gamma \) satisfies a stationary and non-degeneracy conditions with respect to the functional \(\int _{\Gamma } \psi ^\sigma ,\) where \(\sigma = \frac{p+3}{2p}\), we prove the existence of a solution \(u_s\) concentrating along the whole of \(\Gamma \), exponentially small in s at any positive distance from it, provided that s is large and away from certain critical numbers.

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Notes

Acknowledgements

The research of S. Baraket, Z. Khemiri and A. Messaoudi has been supported by the U.R of Nonlinear Analysis and Geometry (Code: UR13ES32), Tunisia. F. Mahmoudi has been supported by Fondecyt Grant 1180526, CONICYT + PIA/Concurso apoyo a Centros Científicos y Tecnológicos de Excelencia, Fondo Basal AFB170001.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Sami Baraket
    • 1
  • Zied  Khemiri
    • 1
  • Fethi Mahmoudi
    • 2
    • 3
  • Abdellatif Messaoudi
    • 4
  1. 1.UR Analyse Non-Linéaire et Géométrie, UR13ES32, Department of Mathematics, Faculty of Sciences of TunisUniversity of Tunis El ManarTunisTunisia
  2. 2.Centro de Modelamiento MatemâticoUniversidad de ChileSantiagoChile
  3. 3.Department of Mathematics, Faculty of Sciences of TunisUniversity of Tunis El ManarTunisTunisia
  4. 4.Institut Préparatoire Aux Etudes d’Ingénieurs- El ManarUniversity of Tunis El ManarTunisTunisia

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