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Weak solution for fractional \((p_1,\ldots , p_m)\)-Laplacian system with Dirichlet-type boundary conditions

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Abstract

In the present paper, we prove the existence and uniqueness of weak solutions to a class of fractional \((p_1,\ldots , p_m)\)-Laplacian problem with Dirichlet-type boundary conditions, the main tool used here is the variational method combined with the theory of fractional Sobolev spaces.

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References

  1. Abassi, A., El Hachimi, A., Jamea, A.: Entropy solutions to nonlinear Neumann problems with $L^1$-data. Int. J. Math. Stat. 2(S08), 4–17 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Ambrosio, V.: Nontrivial solutions for a fractional p-Laplacian problem via Rabier Theorem. Complex Var. Elliptic Equ. 62(6), 838–847 (2017). https://doi.org/10.1080/17476933.2016.1245725

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosio, V., Rǎdulescu, V.D.: Fractional double-phase patterns: concentration and multiplicity of solutions. J. Math. Pures Appl. 142(9), 101–145 (2020)

    Article  MathSciNet  Google Scholar 

  4. Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics, vol. 116, 2nd edn. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  5. Azroul, E., Balaadich, F.: A weak solution to quasilinear elliptic problems with perturbed gradient. Rend. Circ. Mat. Palermo II. Ser. (2020). 10.1007/s12215-020-00488-4

  6. Azroul, E., Balaadich, F.: Existence of solutions for generalized p(x)-Laplacian systems. Rend. Circ. Mat. Palermo II Ser. (2019). 10.1007/s12215-019-00450-z

  7. Azroul, E., Balaadich, F.: Weak solutions for generalized p-Laplacian systems via Young measures. Moroccan J. Pure Appl. Anal. 4(2), 77–84 (2018)

    Article  Google Scholar 

  8. Bogdan, K., Jarohs, S., Kania, E.: Semilinear Dirichlet problem for the fractional Laplacian. Nonlinear Anal. Theory Methods Appl. 193, 111512 (2020)

    Article  MathSciNet  Google Scholar 

  9. Caffarelli, L.: Nonlocal diffusions, drifts and games, Nonlinear partial differential equations, vol. 7, pp. 37–52. Springer, Heidelberg (2012). (Abel Symp.)

    Book  Google Scholar 

  10. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  11. Djida, J.D., Nieto, J.J., Area, I.: Nonlocal time-porous medium equation: weak solutions and finite speed of propagation. Discrete Contin. Dyn. Syst. Ser. B 24, 4031–4053 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  13. Mingqi, X., Rǎdulescu, V.D, Zhang, B.: Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity. Calc. Var. Partial Differ. Equ., 58(2), Paper No. 57 (2019)

  14. Molica Bisci, G., Rǎdulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. With a Foreword by Jean Mawhin, Encyclopedia of Mathematics and Its Applications, vol. 162. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  15. Sabri, A., Jamea, A., Talibi, H.A.: Existence of weak solutions for fractional p-Laplacian problem with Dirichlet-type boundary condition. Discuss. Mathe. Differ. Incl. Control Optim. 39, 69–80 (2019)

    Article  MathSciNet  Google Scholar 

  16. Servadei, R., Valdinoci, E.: Mountain pass solutions for nonlocal elliptic operators. J. Math. Anal. Appl. 389(2), 887–898 (2012)

    Article  MathSciNet  Google Scholar 

  17. Van Groesen, E.W.C.: Variational Methods For Nonlinear Operator Equations, pp. 100–191. Mathematisch Centrum, Amsterdam (1976)

    Google Scholar 

  18. Xiang, M., Zhang, B., Rǎdulescu, V.D.: Superlinear Schrödinger–Kirchhoff type problems involving the fractional p-Laplacian and critical exponent. Adv. Nonlinear Anal. 9(1), 690–709 (2020)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to express their sincere gratitude to the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions, which helped them to improve the paper.

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Authors and Affiliations

  1. Faculté des Sciences, EMAPI, Université Chouaib Doukkali, El Jadida, Morocco

    Abdelali Sabri, Ahmed Jamea & Hamad Talibi Alaoui

  2. Centre Régional des Métiers de l’Education et de Formation Casablanca-Settat, El Jadida, Morocco

    Ahmed Jamea

Authors
  1. Abdelali Sabri
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  2. Ahmed Jamea
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  3. Hamad Talibi Alaoui
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Correspondence to Abdelali Sabri.

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Sabri, A., Jamea, A. & Talibi Alaoui, H. Weak solution for fractional \((p_1,\ldots , p_m)\)-Laplacian system with Dirichlet-type boundary conditions. Rend. Circ. Mat. Palermo, II. Ser 70, 1541–1560 (2021). https://doi.org/10.1007/s12215-020-00573-8

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  • DOI: https://doi.org/10.1007/s12215-020-00573-8

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