Advances in Operator Theory

Eigenvalue problems involving the fractional $p(x)$-Laplacian operator

E. ‎Azroul‎, A. ‎Benkirane‎, and M. ‎Shimi‎

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Abstract

‎ ‎‎In this paper‎, ‎we study a nonlocal eigenvalue problem involving variable exponent growth conditions‎, ‎on a bounded domain $\Omega \subset \mathbb{R}^n$‎. ‎Using adequate variational techniques‎, ‎mainly based on Ekeland's variational principle‎, ‎we establish the existence of a continuous family of eigenvalues lying in a neighborhood at the right of the origin‎.

Article information

Source
Adv. Oper. Theory, Volume 4, Number 2 (2019), 539-555.

Dates
Received: 14 September 2018
Accepted: 18 November 2018
First available in Project Euclid: 1 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1543633243

Digital Object Identifier
doi:10.15352/aot.1809-1420

Mathematical Reviews number (MathSciNet)
MR3883152

Subjects
Primary: 35R11: Fractional partial differential equations
Secondary: 35P30‎ ‎35J20

Keywords
Ekeland's variational principle‎ ‎‎fractional $p(x)$-Laplacian operator ‎fractional Sobolev spaces with variable exponent ‎ ‎‎eigenvalues problem

Citation

‎Azroul‎, E.; ‎Benkirane‎, A.; ‎Shimi‎, M. Eigenvalue problems involving the fractional $p(x)$-Laplacian operator. Adv. Oper. Theory 4 (2019), no. 2, 539--555. doi:10.15352/aot.1809-1420. https://projecteuclid.org/euclid.aot/1543633243


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References

  • D. Applebaum, Lévy processes and stochastic calculus, Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009.
  • A. Bahrouni, Comparison and sub-supersolution principles for the fractional $p(x)$-Laplacian, J. Math. Anal. Appl. 458 (2018), no. 2, 1363–1372.
  • A. Bahrouni and V. Rădulescu, On a 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.
  • G. M. Bisci, V. Rădulescu, and R. Servadi, Variational methods for nonlocal fractional problems. With a foreword by Jean Mawhin, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016.
  • C. Bucur and E. Valdinoci, Nonlocal diffusion and applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016.
  • L. Caffarelli, Nonlocal diffusions, drifts and games, Nonlinear partial differential equations, 37–52, Abel Symp., 7, Springer, Heidelberg, 2012.
  • L. M. Del Pezzo, and J. D. Rossi, Traces for fractional Sobolev spaces with variable exponents, Adv. Oper. Theory 2 (2017), 435–446.
  • E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573.
  • X. L. Fan, Remarks on eigenvalue problems involving the $p(x)$-Laplacian, J. Math. Anal. Appl. 352 (2009), no. 1, 85–98.
  • X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843–1852.
  • X. L. Fan, Q. Zhang, and D. Zhao, Eigenvalues of $p(x)$-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306–317.
  • X. L. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl. 263 (2001), 424–446.
  • U. Kaufmann, J. D. Rossi, and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional $p(x)$-Laplacians, Electron. J. Qual. Theory Differ. Equ. 2017, Paper No. 76, 10 pp.
  • O. Kováčik and J. Rákosník, On spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, Czechoslovak Math. J. 41 (1991), no. 4, 592–618.
  • M. Mihăilescu and V. D. Rădulescu, Continuous spectrum for a class of nonhomogeneous differentials operators, Manuscripta Math. 125 (2008), no. 2, 157–167.
  • M. Mihăilescu and V. D. Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2929–2937.
  • M Mihăilescu and D. Stancu-Dumitru, On an eigenvalue problem involving the $p(x)$-Laplace operator plus a nonlocal term, Differ. Equ. Appl. 1 (2009), no 3, 367–378.
  • P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.
  • V. D. Rădulescu and D. Repovš, Partial differential equations with variable exponents: Variational methods and qualitative analysis, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015.
  • R. Servadei and E. Valdinoci, Variational methods for nonlocal operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 2105–2137.
  • R. Servadei and E. Valdinoci, Mountain pass solutions for nonlocal elliptic operators, J. Math. Anal. Appl. 389 (2012), no. 2, 887–898.
  • R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat. 58 (2014), 133–154.
  • S. Shuzhong, Ekeland's variational principle and the mountain pass lemma, Acta Math. Sinica 1 (1985) 348–355.
  • C. Zhang and X. Zhang, Renormalized solutions for the fractional $p(x)$-Laplacian equation with $L^{1}$ data, arXiv:1708.04481v1.