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Existence Results for A Nonlocal Problem Involving the p(x)-Laplacian

Published: 20 February 2013
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Abstract

In the present paper, we deal with two different existence results of solutions for a nonlocal elliptic Dirichlet boundary value problem involving p(x)-Laplacian. The first one is based on the Brouwer fixed point theorem and the Galerkin method which gives a priori estimate of a nontrivial weak soltion. The second one is based on the variational methods. By using Mountain-Pass theorem, we obtain at least one nontrivial weak soltion.

Published in Pure and Applied Mathematics Journal (Volume 2, Issue 1)
DOI 10.11648/j.pamj.20130201.13
Page(s) 20-27
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2013. Published by Science Publishing Group

Keywords

p(x)-Laplacian; Nonlocal Problem; Fixed Point Theorem; Galerkin Method; Variational Methods Moun-tain-Pass Theorem

References
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[3] B. Cekic, A. V. Kalinin, R. A. Mashiyev, M. Avci, "L^{p(x)}(Ω)-estimates of vector fields and some applica-tions to magnetostatics problems", Journal of Math. Analysis and Appl., vol. 389 (2), pp. 838-851, 2012.
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[18] S. Ogras, R. A. Mashiyev, M. Avci, Z. Yucedag, "Existence of Solutions for a Class of Elliptic Systems in ℝⁿ Involning (p(x),q(x))-Laplacian", Journal of Inequalities and Applica-tions, ID612938, pp.20, 2008.
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    Mustafa Avci. (2013). Existence Results for A Nonlocal Problem Involving the p(x)-Laplacian. Pure and Applied Mathematics Journal, 2(1), 20-27. https://doi.org/10.11648/j.pamj.20130201.13

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    ACS Style

    Mustafa Avci. Existence Results for A Nonlocal Problem Involving the p(x)-Laplacian. Pure Appl. Math. J. 2013, 2(1), 20-27. doi: 10.11648/j.pamj.20130201.13

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    AMA Style

    Mustafa Avci. Existence Results for A Nonlocal Problem Involving the p(x)-Laplacian. Pure Appl Math J. 2013;2(1):20-27. doi: 10.11648/j.pamj.20130201.13

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  • @article{10.11648/j.pamj.20130201.13,
      author = {Mustafa Avci},
      title = {Existence Results for A Nonlocal Problem Involving the p(x)-Laplacian},
      journal = {Pure and Applied Mathematics Journal},
      volume = {2},
      number = {1},
      pages = {20-27},
      doi = {10.11648/j.pamj.20130201.13},
      url = {https://doi.org/10.11648/j.pamj.20130201.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130201.13},
      abstract = {In the present paper, we deal with two different existence results of solutions for a nonlocal elliptic Dirichlet boundary value problem involving p(x)-Laplacian. The first one is based on the Brouwer fixed point theorem and the Galerkin method which gives a priori estimate of a nontrivial weak soltion. The second one is based on the variational methods. By using Mountain-Pass theorem, we obtain at least one nontrivial weak soltion.},
     year = {2013}
    }
    

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    T1  - Existence Results for A Nonlocal Problem Involving the p(x)-Laplacian
    AU  - Mustafa Avci
    Y1  - 2013/02/20
    PY  - 2013
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    DO  - 10.11648/j.pamj.20130201.13
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
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    EP  - 27
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.20130201.13
    AB  - In the present paper, we deal with two different existence results of solutions for a nonlocal elliptic Dirichlet boundary value problem involving p(x)-Laplacian. The first one is based on the Brouwer fixed point theorem and the Galerkin method which gives a priori estimate of a nontrivial weak soltion. The second one is based on the variational methods. By using Mountain-Pass theorem, we obtain at least one nontrivial weak soltion.
    VL  - 2
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics, Dicle University, Diyarbakir, Turkey

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