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

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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.

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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. Jazan Technical College, P.O. Box: 241, Jazan, 45952, Kingdom of Saudi Arabia

    K. B. Ali & M. Hsini

  2. Department of Mathematics, Faculty of Sciences of Tunis, Tunis, Tunisia

    K. B. Ali, M. Hsini & K. Kefi

  3. Community College of Rafha, Northern Border University, Rafha, Kingdom of Saudi Arabia

    K. Kefi

  4. Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam

    N. T. Chung

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  1. K. B. Ali
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  2. M. Hsini
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  3. K. Kefi
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  4. N. T. Chung
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Correspondence to N. T. Chung.

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Communicated by Daniel Aron Alpay.

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Ali, K.B., Hsini, M., Kefi, K. et al. On a Nonlocal Fractional p(., .)-Laplacian Problem with Competing Nonlinearities. Complex Anal. Oper. Theory 13, 1377–1399 (2019). https://doi.org/10.1007/s11785-018-00885-9

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  • DOI: https://doi.org/10.1007/s11785-018-00885-9

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