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Positive solutions of the p-Kirchhoff problem with degenerate and sign-changing nonlocal term

  • Phuong Le
  • Nhat Vy Huynh
  • Vu HoEmail author
Article
  • 27 Downloads

Abstract

We establish the existence and multiplicity of positive solutions of the p-Kirchhoff problem
$$\begin{aligned} {\left\{ \begin{array}{ll} -m\left( \int \limits _\Omega |\nabla u|^p \mathrm{d}x\right) \Delta _p u = f(u) &{}\text { in } \Omega ,\\ u=0 &{}\text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$
where \(p>1\), \(\Omega \) is a smooth bounded domain of \({\mathbb {R}}^N\), and \(f\in C({\mathbb {R}}_0^+)\cap L^1({\mathbb {R}}_0^+)\) is subcritical and positive in a right neighborhood of zero. The main feature of our problem is that \(m:{\mathbb {R}}_0^+\rightarrow {\mathbb {R}}\) may be any continuous function such that the integral of m in each connected component of \(m^{-1}((0,+\infty ))\) is controlled by p, f and \(\Omega \). Therefore, in our paper m may be degenerate, i.e., it could vanish, and sign-changing at any number of different points.

Keywords

Degenerate p-Kirchhoff problems Nonlocal problems Positive solutions Multiplicity results 

Mathematics Subject Classification

35A01 35A15 35B09 35D30 35J92 

Notes

Acknowledgements

We would like to express our sincere gratitude to the anonymous reviewers for their valuable comments and suggestions which helped to improve the quality of this paper.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical EconomicsBanking University of Ho Chi Minh CityHo Chi Minh CityVietnam
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of Science, Vietnam University of Ho Chi Minh CityHo Chi Minh CityVietnam
  3. 3.Department of Fundamental SciencesHo Chi Minh City University of TransportHo Chi Minh CityVietnam
  4. 4.Division of Computational Mathematics and Engineering, Institute for Computational ScienceTon Duc Thang UniversityHo Chi Minh CityVietnam
  5. 5.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam

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