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On a Nonlocal Fractional p(., .)-Laplacian Problem with Competing Nonlinearities

  • K. B. Ali
  • M. Hsini
  • K. Kefi
  • N. T. ChungEmail author
Article
  • 12 Downloads

Abstract

The aim of this paper is to study the existence of nontrivial weak solutions for the problem
$$\begin{aligned} \left\{ \begin{array}{ll} M\left( \int _{\Omega \times \Omega }\frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y)|x-y|^{N+p(x,y)s}}dxdy\right) (\Delta )^s_{p(x,.)}u(x)\\ \quad = \lambda f(x,u) - |u(x)|^{q(x)-2}u(x)\quad \hbox {in}~\Omega , \\ u = 0\quad \hbox {in}~\partial \Omega , \end{array} \right. \end{aligned}$$
where \(\Omega \subset \mathbb R^N\), \(N\ge 2\) is a bounded smooth domain, M and f are two continuous functions and \((\Delta )^s_{p(.,.)}\) is the fractional p(., .)-Laplacian while \(\lambda \) is a positive parameter and \(0<s<1\). Using variational techniques combined with the theory of the generalized Lebesgue Sobolev spaces, we prove some existence and multiplicity results for the problem in an appropriate space of functions.

Keywords

\(p(., .)\)-Fractional Laplacian Kirchhoff type problems Variable exponents Variational methods 

Mathematics Subject Classification

35D05 35J60 35J65 35D30 35J58 

Notes

Acknowledgements

The authors would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.02-2017.04.

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Authors and Affiliations

  1. 1.Jazan Technical CollegeJazanKingdom of Saudi Arabia
  2. 2.Department of MathematicsFaculty of Sciences of TunisTunisTunisia
  3. 3.Community College of RafhaNorthern Border UniversityRafhaKingdom of Saudi Arabia
  4. 4.Department of MathematicsQuang Binh UniversityDong HoiVietnam

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