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On the fractional Lazer–McKenna conjecture with superlinear potential

  • B. AbdellaouiEmail author
  • A. Dieb
  • F. Mahmoudi
Article
  • 70 Downloads

Abstract

The goal of this paper is to study the following non-local superlinear elliptic problem
$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} (-\Delta )^s u=|u|^p-\sigma \phi _1 &{}\hbox {in } \Omega ,\\ u=0 &{}\hbox {in } {\mathbb {R}}^N{\setminus }\Omega ,\\ u>0 &{}\hbox {in }\Omega , \end{array}\right. \end{aligned}$$
where \((-\Delta )^s\) is the fractional Laplace operator, \(\Omega \subset \mathbb R^N\) is an open domain with Lipschitz boundary, \(\sigma >0\), \(p\in (1, 2^*_s-1)\) with \(2^{*}_{s}=\frac{2N}{N-2s}\) and \(\phi _1\) is the first positive eigenfunction of the fractional Laplacian with Dirichlet boundary condition. We prove the non-local version of a conjecture due to Lazer and McKenna (Proc R Soc Edinb Sect A 95:275–283, 1983) by constructing solutions with sharp peaks near local maximum points of \(\phi _1\).

Mathematics Subject Classification

35R11 35A15 35A16 35J61 

Notes

Acknowledgements

The authors would like to express their gratitude to the anonymous referee for his/her comments and suggestions that has improved the last version of the manuscript. Part of this work was carried out while the second author was visiting the Center of Mathematical Modeling (CMM) Universidad de Chile. He is thankful for their kind hospitality. B. Abdellaoui is partially supported by Project MTM2013-40846-P and MTM2016-80474-P, MINECO, Spain. 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-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, Département de MathématiquesUniversité Abou Bakr Belkaïd, TlemcenTlemcenAlgeria
  2. 2.Laboratoire d’Analyse Nonlinéaire et Mathématiques AppliquéesUniversité Abou Bakr Belkaïd, TlemcenTlemcenAlgeria
  3. 3.Département de MathématiquesUniversité Ibn Khaldoun, TiaretTiaretAlgeria
  4. 4.Centro de Modelamiento MatemáticoUniversidad de ChileSantiagoChile
  5. 5.Department of Mathematics, Faculty of Sciences of TunisUniversity of Tunis El ManarTunisTunisia

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