Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T19:13:17.934Z Has data issue: false hasContentIssue false
  • English
  • Français

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A NEUMANN PROBLEM INVOLVING VARIABLE EXPONENT GROWTH CONDITIONS

Published online by Cambridge University Press:  01 September 2008

MARIA-MAGDALENA BOUREANU  and
MIHAI MIHĂILESCU
MARIA-MAGDALENA BOUREANU
Affiliation:
Department of Mathematics, University of Craiova, 200585 Craiova, Romania e-mail: [email protected]
MIHAI MIHĂILESCU*
Affiliation:
Department of Mathematics, Central European University, 1051 Budapest, Hungary e-mail: [email protected]
*
*Corresponding author.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study a non-linear elliptic equation involving p(x)-growth conditions and satisfying a Neumann boundary condition on a bounded domain. For that equation we establish the existence of two solutions using as a main tool an abstract linking argument due to Brézis and Nirenberg.

Keywords

35D0535J6035J7058E0576A02
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 3 , September 2008 , pp. 565 - 574
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Acerbi, E. and Mingione, G., Gradient estimates for the p(x)-Laplacean system, J. Reine Angew. Math. 584 (2005), 117–148.Google Scholar
2.Brézis, H., Analyse fonctionnelle: Théorie, méthodes et applications (Masson, Paris, 1992).Google Scholar
3.Brézis, H. and Nirenberg, L., Remarks on finding critical points, Comm. Pure Appl. Math. 44 (8–9) (1991), 939–963.Google Scholar
4.Chen, Y., Levine, S. and Rao, M., Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 66 (4) (2006), 1383–1406.CrossRefGoogle Scholar
5.Diening, L., Theoretical and numerical results for electrorheological fluids, Ph.D. Thesis (University of Frieburg, Germany, 2002).Google Scholar
6.Edmunds, D. E., Lang, J. and Nekvinda, A., On L p(x) norms, Proc. R. Soc. Lond. Ser. A, 455 (1999), 219–225.Google Scholar
7.Edmunds, D. E. and Rákosník, J., Density of smooth functions in W k,p(x) (Ω), Proc. R. Soc. Lond. Ser. A, 437 (1992), 229–236.Google Scholar
8.Edmunds, D. E. and Rákosník, J., Sobolev embedding with variable exponent, Studia Math. 143 (2000), 267–293.Google Scholar
9.Fan, X., Shen, J. and Zhao, D., Sobolev embedding theorems for spaces W k,p(x) (Ω), J. Math. Anal. Appl. 262 (2001), 749–760.Google Scholar
10.Fan, X. L. and Zhang, Q. H., Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843–1852.CrossRefGoogle Scholar
11.Fan, X., Zhang, Q. and Zhao, D., Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306–317.Google Scholar
12.Fan, X. L. and Zhao, D., On the spaces L p(x)(Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2001), 424–446.Google Scholar
13.Halidias, N. and Le, V. K., Multiple solutions for quasilinear elliptic Neumann problems in Orlicz–Sobolev spaces, Boundary Value Prob., 2005 (3) (2005), 299–306.Google Scholar
14.Halsey, T. C., Electrorheological fluids, Science 258 (1992), 761–766.Google Scholar
15.Kováčik, O. and Rákosník, J., On spaces L p(x) and W 1,p(x), Czechoslovak Math. J. 41 (1991), 592–618.Google Scholar
16.Mihăilescu, M., Elliptic problems in variable exponent spaces, Bull. Austral. Math. Soc. 74 (2006), 197–206.Google Scholar
17.Mihăilescu, M., Existence and multiplicity of solutions for an elliptic equation with p(x)-growth conditions, Glasgow Math. J. 48 (2006), 411–418.Google Scholar
18.Mihăilescu, M., Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, Nonlinear Anal. 67 (2007), 1419–1425.Google Scholar
19.Mihăilescu, M., Pucci, P. and Rădulescu, V., Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I 345 (2007), 561–566.Google Scholar
20.Mihăilescu, M., Pucci, P. and Rădulescu, V., Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), 687–698.CrossRefGoogle Scholar
21.Mihăilescu, M. and Rădulescu, V., A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. A: Math., Phys. Eng. Sci. 462 (2006), 2625–2641.Google Scholar
22.Mihăilescu, M. and Rădulescu, V., On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc. 135 (9) (2007), 2929–2937.Google Scholar
23.Mihăilescu, M. and Rădulescu, V., Continuous spectrum for a class of nonhomogeneous differential operators, Manuscripta Mathematica 125 (2008), 157–167.Google Scholar
24.Musielak, J., Orlicz spaces and modular spaces, Lecture Notes in Mathematics, Vol. 1034 (Springer, Berlin, 1983).Google Scholar
25.Ricceri, B., On three critical points theorem, Arch. Math. (Basel) 75 (2000), 220–226.CrossRefGoogle Scholar
26.Ruzicka, M., Electrorheological fluids: Modelingn and mathematical theory (Springer-Verlag, Berlin, 2002).Google Scholar
27.Zhikov, V., Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. 29 (1987), 33–66.Google Scholar