Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T16:54:03.263Z Has data issue: false hasContentIssue false
  • English
  • Français

Qualitative Analysis of Solutions for a Class of Anisotropic Elliptic Equations with Variable Exponent

Part of: Elliptic equations and systems Linear function spaces and their duals

Published online by Cambridge University Press:  15 December 2015

G. A. Afrouzi ,
M. Mirzapour  and
Vicenţiu D. Rădulescu
G. A. Afrouzi
Affiliation:
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran ([email protected]; [email protected])
M. Mirzapour
Affiliation:
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran ([email protected]; [email protected])
Vicenţiu D. Rădulescu*
Affiliation:
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia ([email protected]) and Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, PO Box 1-764, 014700 Bucharest, Romania
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

We are concerned with the degenerate anisotropic problem

We first establish the existence of an unbounded sequence of weak solutions. We also obtain the existence of a non-trivial weak solution if the nonlinear term f has a special form. The proofs rely on the fountain theorem and Ekeland's variational principle.

Keywords

degenerate anisotropic Sobolev spacesvariable exponentDirichlet boundary value conditionfountain theoremEkeland variational principle

MSC classification

Secondary: 35J62: Quasilinear elliptic equations 35J70: Degenerate elliptic equations 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 541 - 557
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical points theory and applications, J. Funct. Analysis 14 (1973), 349–381.Google Scholar
2. Bartsch, T., Infinitely many solutions of a symmetric Dirichlet problem, Nonlin. Analysis 20 (1993), 1205–1216.Google Scholar
3. Boureanu, M. M., Infinitely many solutions for a class of degenerate anisotropic elliptic problems with variable exponent, Taiwanese J. Math. 15 (2011), 2291–2310.Google Scholar
4. Boureanu, M. M., Pucci, P. and Rădulescu, V., Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent, Complex Var. Elliptic Eqns 56 (2011), 755–767.Google Scholar
5. Chen, Y., Levine, S. and Rao, M., Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 66 (2006), 1383–1406.Google Scholar
6. Diening, L., Theoretical and numerical results for electrorheological fluids, PhD thesis, University of Frieburg (2002).Google Scholar
7. Diening, L., Hästö, P. and Nekvinda, A., Open problems in variable exponent Lebesgue and Sobolev spaces, in Proc. FSDONA04, Milovy, Czech Republic, 2004 (ed. Drábek, P. and Rákosník, J.), pp. 38–58 (Academy of Sciences of the Czech Republic, Prague, 2005).Google Scholar
8. Ekeland, I., On the variational principle, J. Math. Analysis Applic. 47 (1974), 324–353.Google Scholar
9. Esedoglu, S. and Osher, S., Decomposition of images by the anisotropic Rudin–Osher–Fatemi model, Commun. Pure Appl. Math. 57 (2004), 1609–1626.CrossRefGoogle Scholar
10. Fan, X. L., Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients, J. Math. Analysis Applic. 312 (2005), 464–477.CrossRefGoogle Scholar
11. Fan, X. L. and Zhao, D., On the spaces L p(x) and W m,p(x) , J. Math. Analysis Applic. 263 (2001), 424–446.Google Scholar
12. Fragalà, I., Gazzola, F. and Kawohl, B., Existence and nonexistence results for anisotropic quasilinear equations, Annales Inst. H. Poincaré Analyse Non Linéaire 21 (2004), 715–734.Google Scholar
13. Halsey, T. C., Electrorheological fluids, Science 258 (1992), 761–766.Google Scholar
14. Kone, B., Ouaro, S. and Traore, S., Weak solutions for anisotropic nonlinear elliptic equations with variable exponents, Electron. J. Diff. Eqns 2009 (2009), 1–11.Google Scholar
15. Kováčik, O. and Rákosník, J., On spaces L p(x ) and W 1,p(x) , Czech. Math. J. 41 (1991), 592–618.Google Scholar
16. Mihăilescu, M. and Moroşanu, G., Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Applic. Analysis 89 (2010), 257–271.Google Scholar
17. Mihăilescu, M. and Rădulescu, V., On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Am. Math. Soc. 135 (2007), 2929–2937.Google Scholar
18. Mihăilescu, M., Pucci, P. and Rădulescu, V., Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Analysis Applic. 340 (2008), 687–698.Google Scholar
19. Rădulescu, V. and Repovš, D., Partial differential equations with variable exponents: variational methods and qualitative analysis, Monographs and Research Notes in Mathematics (Taylor & Francis/Chapman and Hall/CRC, 2015).Google Scholar
20. Rákosník, J., Some remarks to anisotropic Sobolev spaces, I, Beiträge Analysis 13 (1979), 55–68.Google Scholar
21. Rákosník, J., Some remarks to anisotropic Sobolev spaces, II, Beiträge Analysis 15 (1981), 127–140.Google Scholar
22. Ružička, M., Electrorheological fluids: modeling and mathematical theory (Springer, 2002).Google Scholar
23. Willem, M., Minimax theorems (Birkhäuser, 1996).Google Scholar
24. Zhao, J. F., Structure theory of Banach spaces (Wuhan University Press, 1991).Google Scholar
25. Zhikov, V. V., Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. 29 (1987), 33–66.Google Scholar
26. Zhikov, V. V., On Lavrentiev's phenomenon, Russ. J. Math. Phys. 3 (1995), 249–269.Google Scholar
27. Zhikov, V. V., On some variational problem, Russ. J. Math. Phys. 5 (1997), 105–116.Google Scholar