Abstract and Applied Analysis

Multiple Solutions for a Class of Differential Inclusion System Involving the ( p ( x ) , q ( x ) ) -Laplacian

Bin Ge and Ji-Hong Shen

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Abstract

We consider a differential inclusion system involving the ( p ( x ) , q ( x ) ) -Laplacian with Dirichlet boundary condition on a bounded domain and obtain two nontrivial solutions under appropriate hypotheses. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 971243, 19 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365125164

Digital Object Identifier
doi:10.1155/2012/971243

Mathematical Reviews number (MathSciNet)
MR2935158

Zentralblatt MATH identifier
1250.35094

Citation

Ge, Bin; Shen, Ji-Hong. Multiple Solutions for a Class of Differential Inclusion System Involving the $(p(x),q(x))$ -Laplacian. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 971243, 19 pages. doi:10.1155/2012/971243. https://projecteuclid.org/euclid.aaa/1365125164


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References

  • M. Ružicka, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000.
  • V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory,” Izvestiya, vol. 29, no. 1, pp. 33–66, 1987.
  • G. Dai and R. Hao, “Existence of solutions for a $p(x)$-Kirchhoff-type equation,” Journal of Mathematical Analysis and Applications, vol. 359, no. 1, pp. 275–284, 2009.
  • X. Fan and D. Zhao, “On the spaces ${L}^{p(x)}(\Omega )$ and ${W}^{m,p(x)}(\Omega )$,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 424–446, 2001.
  • X. L. Fan and D. Zhao, “On the generalized Orlicz-sobolev spaces ${W}^{k,p(x)}(\Omega )$,” Journal of Gansu Education College, vol. 12, no. 1, pp. 1–6, 1998.
  • B. Ge, X. Xue, and Q. Zhou, “The existence of radial solutions for differential inclusion problems in ${R}^{N}$ involving the $p(x)$-Laplacian,” Nonlinear Analysis, vol. 73, no. 3, pp. 622–633, 2010.
  • B. Ge and X. Xue, “Multiple solutions for inequality Dirichlet problems by the $p(x)$-Laplacian,” Nonlinear Analysis, vol. 11, no. 4, pp. 3198–3210, 2010.
  • B. Ge, X.-P. Xue, and M.-S. Guo, “Three solutions to inequalities of Dirichlet problem driven by $p(x)$-Laplacian,” Applied Mathematics and Mechanics, vol. 31, no. 10, pp. 1283–1292, 2010.
  • B. Ge, X. Xue, and Q. Zhou, “Existence of at least five solutions for a differential inclusion problem involving the $p(x)$-Laplacian,” Nonlinear Analysis, vol. 12, no. 4, pp. 2304–2318, 2011.
  • B. Ge, Q.-M. Zhou, and X.-P. Xue, “Infinitely many solutions for a differential inclusion problem in ${\mathbb{R}}^{N}$ involving p(x)-Laplacian and oscillatory terms,” Zeitschrift für Angewandte Mathematik und Physik. In press.
  • B. Ge, Q.-M. Zhou, and X.-P. Xue, “Multiplicity of solutions for čommentComment on ref. [11?]: Please update the information of these references [10,11?], if possible. differential inclusion problems in ${\mathbb{R}}^{N}$ involving the p(x)-Laplacian,” Monatshefte fur Mathematik. In press.
  • D. G. de Figueiredo, “Semilinear elliptic systems: a survey of superlinear problems,” Resenhas do Instituto de Matemática e Estatística da Universidade de São Paulo, vol. 2, no. 4, pp. 373–391, 1996.
  • F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1983, A Wiley-Interscience Publication.
  • X. Fan, Q. Zhang, and D. Zhao, “Eigenvalues of $p(x)$-Laplacian Dirichlet problem,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 306–317, 2005.
  • C. Li and C.-L. Tang, “Three solutions for a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian,” Nonlinear Analysis, vol. 69, no. 10, pp. 3322–3329, 2008.
  • A. Kristály, “Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 48, no. 2, pp. 465–477, 2005.
  • L. Boccardo and D. Guedes de Figueiredo, “Some remarks on a system of quasilinear elliptic equations,” Nonlinear Differential Equations and Applications, vol. 9, no. 3, pp. 309–323, 2002.
  • X. Fan, J. Shen, and D. Zhao, “Sobolev embedding theorems for spaces ${W}^{k,p(x)}(\Omega )$,” Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 749–760, 2001.
  • X.-L. Fan and Q.-H. Zhang, “Existence of solutions for $p(x)$-Laplacian Dirichlet problem,” Nonlinear Analysis, vol. 52, no. 8, pp. 1843–1852, 2003.
  • X. Fan and S.-G. Deng, “Remarks on Ricceri's variational principle and applications to the $p(x)$-Laplacian equations,” Nonlinear Analysis, vol. 67, no. 11, pp. 3064–3075, 2007.
  • K.C. Chang, Critical Point Theory and Applications, Shanghai Scientific and Technology Prees, Shanghai, China, 1996.
  • N. C. Kourogenis and N. S. Papageorgiou, “Nonsmooth critical point theory and nonlinear elliptic equations at resonance,” Journal of the Australian Mathematical Society. Series A, vol. 69, no. 2, pp. 245–271, 2000.
  • K. C. Chang, “Variational methods for nondifferentiable functionals and their applications to partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 80, no. 1, pp. 102–129, 1981.