Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2015, Article ID 238261, 7 pages
http://dx.doi.org/10.1155/2015/238261
Research Article

On -Anisotropic Problems with Neumann Boundary Conditions

Department of Mathematics (ENSAH), Nonlinear Analysis Laboratory (FSO), University of Mohammed First, 60000 Oujda, Morocco

Received 6 July 2015; Revised 8 November 2015; Accepted 23 November 2015

Academic Editor: Emmanuel Hebey

Copyright © 2015 Anass Ourraoui. 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.

Linked References

  1. C. O. Alves and A. El Hamidi, “Existence of solution for an anisotropic equation with critical exponent,” Nonlinear Analysis: Theory, Methods & Applications, vol. 4, pp. 611–624, 2005. View at Google Scholar
  2. M.-M. Boureanu, P. Pucci, and V. D. Radulescu, “Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent,” Complex Variables and Elliptic Equations, vol. 56, no. 7–9, pp. 755–767, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. M.-M. Boureanu, “Critical point methods in degenerate anisotropic problems with variable exponent,” Studia Universitatis Babes-Bolyai, Mathematica, vol. 55, no. 4, pp. 27–39, 2010. View at Google Scholar
  4. N. T. Chung and H. Q. Toan, “On a class of anisotropic elliptic equations without Ambrosetti-Rabinowitz type conditions,” Nonlinear Analysis. Real World Applications, vol. 16, pp. 132–145, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. F. C. Cîrstea and J. Vétois, “Fundamental solutions for anisotropic elliptic equations: existence and a priori estimates,” Communications in Partial Differential Equations, vol. 40, no. 4, pp. 727–765, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. D. S. Dumitru, “Multiplicity of solutions for a nonlinear degenerate problem in anisotropic variable exponent spaces,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 36, no. 1, pp. 117–130, 2013. View at Google Scholar · View at MathSciNet
  7. A. El Hamidi and J. Vétois, “Sharp Sobolev asymptotics for critical anisotropic equations,” Archive for Rational Mechanics and Analysis, vol. 192, no. 1, pp. 1–36, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. X. Fan, “Anisotropic variable exponent Sobolev spaces and p(x)-Laplacian equations,” Complex Variables and Elliptic Equations, vol. 56, no. 7-9, pp. 623–642, 2011. View at Publisher · View at Google Scholar
  9. I. Fragalà, F. Gazzola, and B. Kawohl, “Existence and nonexistence results for anisotropic quasilinear elliptic equations,” Annales de l'Institut Henri Poincaré C: Non Linear Analysis, vol. 21, no. 5, pp. 715–734, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. B. Kone, S. Ouaro, and S. Traore, “Weak solutions for anisotropic nonlinear elliptic equations with variable exponents,” Electronic Journal of Differential Equations, vol. 2009, no. 144, pp. 1–11, 2009. View at Google Scholar · View at MathSciNet
  11. A. Ourraoui, “On a nonlocal p(x)-Laplacian equations via genus theory,” Rivista di Matematica della Università di Parma, In press.
  12. J. Vétois, “Strong maximum principles for anisotropic elliptic and parabolic equations,” Advanced Nonlinear Studies, vol. 12, no. 1, pp. 101–114, 2012. View at Google Scholar · View at MathSciNet · View at Scopus
  13. J. Vétoiss, “Existence and regularity for critical Anisotropic equations with critical directions,” Advances in Differential Equations, vol. 16, no. 1-2, pp. 61–83, 2011. View at Google Scholar · View at MathSciNet · View at Scopus
  14. J. Batt and Y. Li, “The positive solutions of the Matukuma equation and the problem of finite radius and finite mass,” Archive for Rational Mechanics and Analysis, vol. 198, no. 2, pp. 613–675, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. A. J. Simmonds, “Electro-rheological valves in a hydraulic circuit,” IEE Proceedings D: Control Theory and Applications, vol. 138, no. 4, pp. 400–404, 1991. View at Publisher · View at Google Scholar · View at Scopus
  16. S. N. Antontsev and J. F. Rodrigues, “On stationary thermorheological viscous flows,” Annali dell'Università di Ferrara Sezione VII. Scienze Matematiche, vol. 52, pp. 19–36, 2006. View at Google Scholar
  17. V. V. Zhikov, “Averaging of functionals in the calculus of variations and elasticity,” Mathematics of the USSR-Izvestiya, vol. 29, no. 1, pp. 33–66, 1987. View at Publisher · View at Google Scholar
  18. Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. M. Bendahmane, M. Langlais, and M. Saad, “On some anisotropic reaction-diffusion systems with L1-data modeling the propagation of an epidemic disease,” Nonlinear Analysis: Theory, Methods & Applications, vol. 54, no. 4, pp. 617–636, 2003. View at Publisher · View at Google Scholar
  20. D. E. Edmunds and J. R'akosnk, “Sobolev embedding with variable exponent,” Studia Mathematica, vol. 143, no. 3, pp. 267–293, 2000. View at Google Scholar · View at MathSciNet · View at Scopus
  21. O. Kováčik and J. Rákosník, “On spaces Lp(x) and Wk,p(x),” Czechoslovak Mathematical Journal, vol. 41, pp. 592–618, 1991. View at Google Scholar
  22. D. Geng, “Infinitely many solutions of p-Laplacian equations with limit subcritical growth,” Applied Mathematics and Mechanics, vol. 28, no. 10, pp. 1373–1382, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. P. Pucci, “Geometric description of the mountain pass critical points,” in Contemporary Mathematicians, vol. 2, pp. 469–471, Birkhuser, Basel, Switzerland, 2014. View at Google Scholar
  24. P. Pucci and V. Radulescu, “The impact of the mountain pass theory in nonlinear analysis: a mathematical survey,” Bollettino dell'Unione Matematica Italiana Series IX, vol. 3, pp. 543–584, 2010. View at Google Scholar
  25. M. Willem, Minimax Theorems, Birkhuser, 1996.