• Views 439
• Citations 0
• ePub 34
• PDF 356
`International Journal of Partial Differential EquationsVolume 2013 (2013), Article ID 364251, 7 pageshttp://dx.doi.org/10.1155/2013/364251`
Research Article

## Solutions of Nonlocal -Laplacian Equations

1Faculty of Economics and Administrative Sciences, Batman University, 72000 Batman, Turkey
2Faculty of Education, Bayburt University, 69000 Bayburt, Turkey

Received 5 March 2013; Accepted 10 September 2013

Copyright © 2013 Mustafa Avci and Rabil Ayazoglu (Mashiyev). This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883.
2. M. Avci, B. Cekic, and R. A. Mashiyev, “Existence and multiplicity of the solutions of the p(x)-Kirchhoff type equation via genus theory,” Mathematical Methods in the Applied Sciences, vol. 34, no. 14, pp. 1751–1759, 2011.
3. G. Dai and R. Hao, “Existence of solutions for a p(x)-Kirchho¤-type equation,” Journal of Mathematical Analysis and Applications, vol. 359, pp. 275–284, 2009.
4. X. Fan, “On nonlocal p(x)-Laplacian Dirichlet problems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 72, no. 7-8, pp. 3314–3323, 2010.
5. M. Avci, “Existence and multiplicity of solutions for Dirichlet problems involving the p(x)-Laplace operator,” Electronic Journal of Differential Equations, vol. 2013, no. 14, pp. 1–9, 2013.
6. B. Cekic and R. A. Mashiyev, “Existence and localization results for p(x)-Laplacian via Topological Methods,” Fixed Point Theory and Applications, vol. 2010, Article ID 120646, 7 pages, 2010.
7. X.-L. Fan and Q.-H. Zhang, “Existence of solutions for p(x)-Laplacian Dirichlet problem,” Nonlinear Analysis: Theory, Methods and Applications, vol. 52, no. 8, pp. 1843–1852, 2003.
8. X. Fan, “Eigenvalues of the p(x)-Laplacian Neumann problems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 67, no. 10, pp. 2982–2992, 2007.
9. M. Mihailescu and V. Radulescu, “On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent,” Proceedings of the American Mathematical Society, vol. 135, no. 9, pp. 2929–2937, 2007.
10. S. N. Antontsev and S. I. Shmarev, “A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions,” Nonlinear Analysis: Theory, Methods and Applications, vol. 60, no. 3, pp. 515–545, 2005.
11. S. N. Antontsev and J. F. Rodrigues, “On stationary thermo-rheological viscous flows,” Annali Dell' Università Di Ferrara, vol. 52, no. 1, pp. 19–36, 2006.
12. T. C. Halsey, “Electrorheological fluids,” Science, vol. 258, no. 5083, pp. 761–766, 1992.
13. M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000.
14. V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory,” Mathematics of the USSR-Izvestiya, vol. 9, pp. 33–66, 1987.
15. D. Liu, X. Wang, and J. Jinghua Yao, “On $\left({p}_{1}\left(x\right),{p}_{2}\left(x\right)\right)$-Laplace equations,” http://arxiv.org/pdf/1205.1854.pdf.
16. M.-M. Boureanua, P. Pucci, and V. D. Rǎdulescu, “Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponenty,” Complex Variables and Elliptic Equations, vol. 56, no. 7–9, pp. 755–767, 2011.
17. X. Fan, J. Shen, and D. Zhao, “Sobolev embedding theorems for spaces Wk,p(x)(Ω),” Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 749–760, 2001.
18. X. L. Fan and D. Zhao, “On the spaces Lp(x) (Ω) and Wk,p(x) (Ω),” Journal of Mathematical Analysis and Applications, vol. 263, pp. 424–446, 2001.
19. O. Kovacik and J. Rakosnik, “On spaces Lp(x) and Wk,p(x),” Czechoslovak Mathematical Journal, vol. 41, no. 116, pp. 592–618, 1991.
20. K. C. Chang, Critical Point Theory and Applications, Shanghai Scientific and Technology Press, Shanghai, China, 1986.
21. M. Willem, Minimax Theorems, Birkhauser, Basel, Switzerland, 1996.