Existence and multiplicity results for fractional p(x)-Laplacian Dirichlet problem

O. Chakrone; K. Belhadj; M. R. Sidi Ammi; A. Zerouali

doi:10.2478/mjpaa-2022-0011

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Existence and multiplicity results for fractional p(x)-Laplacian Dirichlet problem

Moroccan Journal of Pure and Applied Analysis

Volume 8 (2022): Issue 2 (May 2022)

By: O. Chakrone, K. Belhadj, M. R. Sidi Ammi and A. Zerouali

Open Access

|May 2022

Abstract

In this paper, we study a class of fractional p(x)-Laplacian Dirichlet problems in a bounded domain with Lipschitz boundary. Using variational methods, we prove in different situations the existence and multiplicity of solutions.

References

[1] M. Ait Hammou, Nonlinear problem involving the fractional p(x)-Laplacian operator by topological degree. Arxiv.
Search in Google Scholar Back to article
[2] E. Azroul, A. Benkirane and M. Shimi, Eigenvalue problems involving the fractional p(x)-Laplacian operator, Adv. Oper. Theory, Volume 4, Number 2 (2019), 539–555.
Search in Google Scholar Back to article
[3] A. Baalal and M. Berghout, Traces and fractional Sobolev extension domains with variable exponent, Int. J. Math. Anal. (Ruse) 12(2018), no. 2, 85–98.
Search in Google Scholar Back to article
[4] A. Bahrouni, Comparison and sub-supersolution principles for the fractional p(x)-Laplacian, J. Math. Anal. Appl. 458(2018), no 2, 1363–372.10.1016/j.jmaa.2017.10.025
Search in Google Scholar Back to article
[5] A. Bahrouni and V. D. Radulescu, On new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. Ser.S 11(2018), no. 3, 379–389.
Search in Google Scholar Back to article
[6] ] B. Barrios, E. Colorado, A. De Pablo, U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differ. Equ., 252 (2012), 6133-6162.10.1016/j.jde.2012.02.023
Search in Google Scholar Back to article
[7] M.M. Boureanu and D. N. Udrea, Existence and multiplicity results for elliptic problems with p(.)-growth conditions, Nonlinear Analysis: Real World Applications 14, 2013, pp. 1829–1844.10.1016/j.nonrwa.2012.12.001
Search in Google Scholar Back to article
[8] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differ. Equ., 32 (2007), 1245-1260.10.1080/03605300600987306
Search in Google Scholar Back to article
[9] H. Chen, H. Hajaiej, Y. Wang, On a class of semilinear fractional elliptic equations involving outside Dirac data, Nonlinear Anal. (TMA), 125 (2015), 639-658.10.1016/j.na.2015.06.002
Search in Google Scholar Back to article
[10] D.G. Costa, An invariation to variationnal methods in differential equation; Birkhauser. Boston 2007.10.1007/978-0-8176-4536-6
Search in Google Scholar Back to article
[11] P. Drabek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A. 127, pp. 703–726, 1997.10.1017/S0308210500023787
Search in Google Scholar Back to article
[12] D. E. Edmunds and J. Rakosnik, Sobolev embeddings with variable exponent, Studia Math. 143, pp. 267–293, 2000.10.4064/sm-143-3-267-293
Search in Google Scholar Back to article
[13] X.L. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446.10.1006/jmaa.2000.7617
Search in Google Scholar Back to article
[14] Y. Jabri, The mountain Pass Theorem. Variants, Generalizations and Some Application. Cambridge University Press, 2003.10.1017/CBO9780511546655
Search in Google Scholar Back to article
[15] R.Kajikia, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal. 225(2005) 352–370.
Search in Google Scholar Back to article
[16] B. Karim. A. Zerouali, O. Chakrone, Existence and multiplicity results for Steklov problem with p(.)-growth conditions, Bulletin of the Iranian Mathematical Society(2018)44:819–836.10.1007/s41980-018-0054-5
Search in Google Scholar Back to article
[17] U. Kaufman, J. D. Rossi and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians, Electro. J. Qual.Theory Differ. Equ. 76, pp.1–10, 2017.10.14232/ejqtde.2017.1.76
Search in Google Scholar Back to article
[18] O. Kovácik, J. Rákosnik, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41(1991), 592–618.10.21136/CMJ.1991.102493
Search in Google Scholar Back to article
[19] S. Mosconi, K. Perera, M. Squassina, Y. Yang, A Brezis-Nirenberg result for the fractional p-Laplacian, Calc. Var. Partial Differ. Equ., 55 (2016), 55:105.10.1007/s00526-016-1035-2
Search in Google Scholar Back to article
[20] Nguyen Thanh Chung, Multiple solutions for a class of p(x)-Laplacian problems involving concave-convex nonlinearities, Electronic Journal of Qualitative Theory Equations 2013, No. 26, 1–17.
Search in Google Scholar Back to article
[21] L.M. Pezzo and J.D. Rossi, Trace for fractional Sobolev spaces with variable exponents, Adv. Oper. Theory 2, pp.435–446, 2017.
Search in Google Scholar Back to article
[22] M. Xiang, B. Zhang and D. Yang, Multiplicity results for variable-order fractional Laplacian equations with variable growth, Nonlinear Anal.178(2019), 190–204.10.1016/j.na.2018.07.016
Search in Google Scholar Back to article
[23] C. Zhang and X. Zhang, Renormalized solutions for the fractional p(x)-Laplacian equation with L1 data, arXiv:1708, 04481v1.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/mjpaa-2022-0011 | Journal eISSN: 2351-8227

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Language: English

Page range: 148 - 162

Submitted on: Dec 8, 2020

Accepted on: Dec 23, 2021

Published on: May 28, 2022

Published by: Sciendo

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

Critical point theorem,

Fractional ()-Laplacian,

fractional Sobolev space,

Related subjects:

Differential equations and dynamical systems,

Probability and statistics,

Numerical and computational mathematics

© 2022 O. Chakrone, K. Belhadj, M. R. Sidi Ammi, A. Zerouali, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 8 (2022): Issue 2 (May 2022)