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

Mustafa Avci

doi:doi:10.11648/j.pamj.20130201.13

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

Mustafa Avci

Published in Pure and Applied Mathematics Journal (Volume 2, Issue 1)

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

[1]	S. N. Antontsev, J. F. Rodrigues, "On stationary ther-mo-rheological viscous flows", Ann. Univ. Ferrara, Sez. 7, Sci. Mat., vol.52, pp. 19-36, 2006.
[2]	M. Avci , B. Cekic, R. A. Mashiyev , "Existence and multip-licity of the solutions of the p(x)-Kirchhoff type equation via genus theory", Math. Methods in the Applied Sciences, vol. 34 (14) , pp.1751-1759, 2011.
[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.
[4]	F. J. S. A. Corrêa, G. M. Figueiredo, "On a elliptic equation of p-kirchhoff type via variational methods", Bull. Austral. Math. Soc., vol. 74, pp.263-277, 2006.
[5]	G. Dai, D. Liu, "Infinitely many positive solutions for a p(x)-Kirchhoff-type equation", J. Math. Anal. Appl., vol.359 (2), pp.704-710, 2009.
[6]	G. Dai, R. Hao, "Existence of solutions for a p(x)-Kirchhoff-type equation", J. Math. Anal. Appl., vol. 359 (2), pp. 275-284, 2009.
[7]	X. L. Fan., "On nonlocal p(x)-Laplacian Dirichlet problems", Nonlinear Anal. vol.72, pp. 3314-3323, 2010.
[8]	X. L. Fan, D. Zhao, "On the spaces L^{p(x)}and W^{m,p(x)}", J. Math. Anal. Appl., vol. 263, pp. 424-446, 2001.
[9]	X. Fan, Q. H. Zhang, "Existence of solutions for p(x)-Laplacian Dirichlet problems", Nonlinear Anal., vol. 52, pp. 1843-1852, 2003.
[10]	P. A. Hästö, "On the variable exponent Dirichlet energy integral", Comm. Pure Appl. Anal., vol. 5 (3), pp. 413-420, 2006.
[11]	X. He, W. Zou, "Infinitely many positive solutions for Kir-chhoff-type problems", Nonlinear Anal. vol. 70, pp. 1407-1414, 2009.
[12]	S. Kesavan, Topics in Functional Analysis and Applications, Wiley, NewYork, 1989.
[13]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[14]	O. Kovăčik, J. Răkosnik, "On spaces L^{p(x)} and W^{1,p(x)}", Czechoslovak Math. J. vol. 41 (116), pp. 592-618, 1991.
[15]	T. F. Ma, "Remarks on an elliptic equation of Kirchhoff type", Nonlinear Anal., vol. 63, pp. 1967-1977, 2005.
[16]	H. Ma, G. Dai, "Existence results for a variable exponent elliptic problem via topological method", Boundary Value Problems, vol. 2012:99. doi:10.1186/1687-2770-2012-99.
[17]	M. Mihăilescu, V. Rădulescu, "On a nonhomogeneous qua-silinear eigenvalue problem in Sobolev spaces with variable exponent", Proceedings Amer. Math. Soc., vol. 135 (9), pp. 2929-2937, 2007.
[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.
[19]	M. Růžička, Electrorheological fluids: Modeling and Ma-thematical Theory, Lecture Notes in Mathematics, Sprin-ger-Verlag: Berlin, 2000.
[20]	M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
[21]	F. Wang, Y. An, "Existence of Nontrivial Solution for a Nonlocal Elliptic Equation with Nonlinear Boundary Condi-tion", Boundary Value Problems Vol. 2009, Article ID 540360, 8 pages, doi:10.1155/2009/540360.
[22]	V.V. Zhikov, "Averaging of functionals of the calculus of variations and elasticity theory", Math. USSR. Izv., vol. 9, pp. 33-66, 1987

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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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TY  - JOUR
T1  - Existence Results for A Nonlocal Problem Involving the p(x)-Laplacian
AU  - Mustafa Avci
Y1  - 2013/02/20
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130201.13
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
SP  - 20
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

Mustafa Avci

Department of Mathematics, Dicle University, Diyarbakir, Turkey

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Plain Text BibTeX RIS

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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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Existence Results for A Nonlocal Problem Involving the p(x)-Laplacian
AU  - Mustafa Avci
Y1  - 2013/02/20
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130201.13
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
SP  - 20
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  -

Copy | Download