Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio


IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2017: 0.23

Online
ISSN
1572-9176
See all formats and pricing
More options …
Ahead of print

Issues

p(x)-biharmonic operator involving the p(x)-Hardy inequality

Abdelouahed El Khalil
  • Department of Mathematics and Statistics, College of Science, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, 11623 Riyadh, Saudi Arabia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Mostafa El Moumni
  • Department of Mathematics, Faculty of Sciences El Jadida, University Chouaib Doukkali, P.O. Box 20, 24000 El Jadida, Morocco
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Moulay Driss Morchid Alaoui
  • Corresponding author
  • Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, P.O. Box 1796 Fez, Morocco
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Abdelfattah Touzani
  • Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, P.O. Box 1796 Fez, Morocco
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-03-23 | DOI: https://doi.org/10.1515/gmj-2018-0013

Abstract

In this work, we investigate the spectrum denoted by Λ for the p(x)-biharmonic operator involving the Hardy term. We prove the existence of at least one non-decreasing sequence of positive eigenvalues of this problem such that supΛ=+. Moreover, we prove that infΛ>0 if and only if the domain Ω satisfies the p(x)-Hardy inequality.

Keywords: nonlinear eigenvalue problems; variational methods,Ljusternik–Schnirelman theory

MSC 2010: 58E05; 35J35; 35J60; 47J10

References

  • [1]

    A. Ancona, On strong barriers and an inequality of Hardy for domains in 𝐑n, J. Lond. Math. Soc. (2) 34 (1986), no. 2, 274–290. Google Scholar

  • [2]

    S. N. Antontsev and S. I. Shmarev, On the localization of solutions of elliptic equations with nonhomogeneous anisotropic degeneration (in Russian), Sibirsk. Mat. Zh. 46 (2005), no. 5, 963–984; tanslation in Siberian Math. J. 46 (2005), no. 5, 765–782. Google Scholar

  • [3]

    Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406. CrossrefGoogle Scholar

  • [4]

    F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (2011), no. 17, 5962–5974. Web of ScienceGoogle Scholar

  • [5]

    E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in Lp(Ω), Math. Z. 227 (1998), no. 3, 511–523. Google Scholar

  • [6]

    L. Diening, P. Harjulehto, P. Hästö and M. Růz̆ic̆ka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011. Web of ScienceGoogle Scholar

  • [7]

    J. W. Dold, V. A. Galaktionov, A. A. Lacey and J. L. Vázquez, Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 26 (1998), no. 4, 663–687. Google Scholar

  • [8]

    A. El Khalil, M. D. Morchid Alaoui and A. Touzani, On the spectrum of the p-biharmonic operator involving p-Hardy’s inequality, Appl. Math. (Warsaw) 41 (2014), no. 2–3, 239–246. CrossrefGoogle Scholar

  • [9]

    X. L. Fan and X. Fan, A Knobloch-type result for p(t)-Laplacian systems, J. Math. Anal. Appl. 282 (2003), no. 2, 453–464. Google Scholar

  • [10]

    X. L. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), no. 2, 424–446. Web of ScienceGoogle Scholar

  • [11]

    A. Ferrero and G. Warnault, On solutions of second and fourth order elliptic equations with power-type nonlinearities, Nonlinear Anal. 70 (2009), no. 8, 2889–2902. CrossrefWeb of ScienceGoogle Scholar

  • [12]

    A. Kufner, Weighted Sobolev Spaces, John Wiley & Sons, New York, 1985. Google Scholar

  • [13]

    A. Kufner and B. Opic, Hardy-Type Inequalities, Pitman Res. Notes Math. Ser. 219, Longman Scientific & Technical, Harlow, 1990. Google Scholar

  • [14]

    J. L. Lewis, Uniformly fat sets, Trans. Amer. Math. Soc. 308 (1988), no. 1, 177–196. CrossrefGoogle Scholar

  • [15]

    T. C. Halsey, Electrorheological fluids, Science 258 (1992), no. 5083, 761–766. CrossrefGoogle Scholar

  • [16]

    G. H. Hardy, An inequality between integrals, Messenger Math. 54 (1925), 150–156. Google Scholar

  • [17]

    P. Harjulehto, P. Hästö, V. Lê, Út and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), no. 12, 4551–4574. CrossrefWeb of ScienceGoogle Scholar

  • [18]

    V. G. Maz’ya, Sobolev Spaces, Springer Ser. Soviet Math., Springer, Berlin, 1985. Google Scholar

  • [19]

    M. Mihăilescu, Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, Nonlinear Anal. 67 (2007), no. 5, 1419–1425. Google Scholar

  • [20]

    É. Mitidieri, A simple approach to Hardy inequalities (in Russian), Mat. Zametki 67 (2000), no. 4, 563–572; translation in Math. Notes 67 (2000), no. 3-4, 479–486. Google Scholar

  • [21]

    T. G. Myers, Thin films with high surface tension, SIAM Rev. 40 (1998), no. 3, 441–462. CrossrefGoogle Scholar

  • [22]

    J. Necás, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Sc. Norm. Super. Pisa (3) 16 (1962), 305–326. Google Scholar

  • [23]

    M. P. Owen, The Hardy–Rellich inequality for polyharmonic operators, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 825–839. CrossrefGoogle Scholar

  • [24]

    F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung, Proceedings of the International Congress of Mathematicians 1954. Volume 3, North-Holland, Amsterdam (1956), 243–250. Google Scholar

  • [25]

    M. Růz̆ic̆ka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. 1748, Springer, Berlin, 2000. Google Scholar

  • [26]

    A. Szulkin, Ljusternik–Schnirelmann theory on 𝐶1-manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 2, 119–139. Google Scholar

  • [27]

    A. Wannebo, Hardy inequalities, Proc. Amer. Math. Soc. 109 (1990), no. 1, 85–95. CrossrefGoogle Scholar

  • [28]

    J. H. Yao, Solutions for Neumann boundary value problems involving p(x)-Laplace operators, Nonlinear Anal. 68 (2008), no. 5, 1271–1283. Web of ScienceGoogle Scholar

  • [29]

    A. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue–Sobolev spaces, Nonlinear Anal. 69 (2008), no. 10, 3629–3636. Web of ScienceCrossrefGoogle Scholar

  • [30]

    E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear monotone operators, Springer, New York, 1990. Google Scholar

  • [31]

    V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710. Google Scholar

About the article

Received: 2015-10-11

Revised: 2016-12-09

Accepted: 2017-09-22

Published Online: 2018-03-23


Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2018-0013.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in