Advances in Operator Theory

Multiplicity of solutions for a class of Neumann elliptic systems in anisotropic Sobolev spaces with variable exponent

Mohamed Saad Bouh Elemine Vall and Ahmed Ahmed

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Abstract

‎‎In this paper‎, ‎we prove the existence of infinitely many solutions of a system of boundary value problems involving flux boundary conditions in anisotropic variable exponent Sobolev spaces‎, ‎by applying a critical point variational principle obtained by Ricceri as a consequence of a more general variational principle and the theory of the anisotropic variable exponent Sobolev spaces‎.

Article information

Source
Adv. Oper. Theory, Volume 4, Number 2 (2019), 497-513.

Dates
Received: 24 August 2018
Accepted: 2 November 2018
First available in Project Euclid: 1 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1543633240

Digital Object Identifier
doi:10.15352/aot.1808-1409

Mathematical Reviews number (MathSciNet)
MR3883149

Zentralblatt MATH identifier
07009322

Subjects
Primary: 34A34: Nonlinear equations and systems, general
Secondary: 35D30‎ ‎35J50

Keywords
Neumann elliptic problem‎‎ gradient system‎ ‎ ‎‎‎weak solution ‎ variational principle‎ ‎ anisotropic variable exponent Sobolev space

Citation

Elemine Vall, Mohamed Saad Bouh; Ahmed, Ahmed. Multiplicity of solutions for a class of Neumann elliptic systems in anisotropic Sobolev spaces with variable exponent. Adv. Oper. Theory 4 (2019), no. 2, 497--513. doi:10.15352/aot.1808-1409. https://projecteuclid.org/euclid.aot/1543633240


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References

  • A. Ahmed, H. Hjiaj, and A. Touzani, Existence of infinitely many weak solutions for a Neumann elliptic equations involving the $\vec{p}(x)$-laplacian operator, Rend. Circ. Mat. Palermo (2) 64 (2015), no. 3, 459–473.
  • A. Bechah, Local and global estimates for solutions of systems involving the $p$-Laplacian in unbounded domains, Electron. J. Differential Equations 2001, No. 19, 14 pp.
  • M. Bendahmane, M. Chrif, and S. El Manouni, An approximation result in generalized anisotropic Sobolev spaces and applications, Z. Anal. Anwend., 30 (2011), no. 3, 341–353.
  • M. Bendahmane and F. Mokhtari, Nonlinear elliptic systems with variable exponents and measure data, Moroccan J. Pure Appl. Anal. 1 (2015), no. 2, Article ID 8, 108–125.
  • A. Bensedik and M. Bouchekif, On certain nonlinear elliptic systems with indefinite terms, Electron. J. Differential Equations2002, No. 83, 16 pp.
  • L. Boccardo and D. Guedes De Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 9 (2002), 309–323.
  • M.-M. Boureanu, Infinitely many solutions for a class of degenerate anisotropic elliptic problems with variable exponent, Taiwanese J. Math. 15 (2011), no. 5, 2291–2310.
  • M.-M. Boureanu and V. D. Radulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent, Nonlinear Anal. 75 (2012), no. 12, 4471–4482. 4471–4482.
  • M.-M Boureanu, C. Udrea, and D.-N. Udrea, Anisotropic problems with variable exponents and constant Dirichlet condition, Electron. J. Differential Equations 2013, No. 220, 13 pp.
  • \labellivD.V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue spaces. Foundations and harmonic analysis, Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Heidelberg, 2013.
  • \labellp L. Diening, P. Harjulehto, P. Hästä, and M. R\ružička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017. Springer, Heidelberg, 2011.
  • X. L. Fan and D. Zhao, On the generalised Orlicz-Sobolev space $W^{k,p(x)}(\Omega)$, J. Gansu Educ. College 12 (1998), no. 1, 1–6.
  • B. Kone, S. Ouaro, and S. Traore, Weak solutions for anisotropic nonlinear elliptic equations with variable exponents, Electron. J. Differential Equations 2009, No. 144, 11 pp.
  • M. Mih$\check{a}$ilescu and G. Morosanu, Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Appl. Anal. 89 (2010), no. 2, 257–271.
  • D. S. Moschetto, Infinitely many solutions to the Dirichlet problem for quasilinear elliptic systems involving the $p(x)$ and $q(x)$-Laplacian, Matematiche (Catania) 63 (2008), no. 1, 223–233.
  • D. S. Moschetto, Infinitely many solutions to the Neumann problem for quasilinear elliptic systems involving the $p(x)$ and $q(x)$-Laplacian, Int. Math. Forum 4 (2009), no. 21-24, 1201–1211.
  • B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), no. 1-2, 401–410.
  • D. Terman, Radial solutions of an elliptic system: solutions with a prescribed winding number, Houston J. Math. 15 (1989), no. 3, 425–458.