EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A NEUMANN PROBLEM INVOLVING VARIABLE EXPONENT GROWTH CONDITIONS
Glasgow mathematical journal, Tome 50 (2008) no. 3, pp. 565-574

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.1017/S0017089508004424
Mots-clés : 35D05, 35J60, 35J70, 58E05, 76A02
BOUREANU, MARIA-MAGDALENA; MIHĂILESCU, MIHAI. EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A NEUMANN PROBLEM INVOLVING VARIABLE EXPONENT GROWTH CONDITIONS. Glasgow mathematical journal, Tome 50 (2008) no. 3, pp. 565-574. doi: 10.1017/S0017089508004424
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[1] 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] 2.Brézis, H., Analyse fonctionnelle: Théorie, méthodes et applications (Masson, Paris, 1992). Google Scholar

[3] 3.Brézis, H. and Nirenberg, L., Remarks on finding critical points, Comm. Pure Appl. Math. 44 (8–9) (1991), 939–963. Google Scholar

[4] 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. Google Scholar | DOI

[5] 5.Diening, L., Theoretical and numerical results for electrorheological fluids, Ph.D. Thesis (University of Frieburg, Germany, 2002). Google Scholar

[6] 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] 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] 8.Edmunds, D. E. and Rákosník, J., Sobolev embedding with variable exponent, Studia Math. 143 (2000), 267–293. Google Scholar

[9] 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] 10.Fan, X. L. and Zhang, Q. H., Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843–1852. Google Scholar | DOI

[11] 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] 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] 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] 14.Halsey, T. C., Electrorheological fluids, Science 258 (1992), 761–766. Google Scholar

[15] 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] 16.Mihăilescu, M., Elliptic problems in variable exponent spaces, Bull. Austral. Math. Soc. 74 (2006), 197–206. Google Scholar

[17] 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] 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] 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] 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. Google Scholar | DOI

[21] 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] 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] 23.Mihăilescu, M. and Rădulescu, V., Continuous spectrum for a class of nonhomogeneous differential operators, Manuscripta Mathematica 125 (2008), 157–167. Google Scholar

[24] 24.Musielak, J., Orlicz spaces and modular spaces, Lecture Notes in Mathematics, Vol. 1034 (Springer, Berlin, 1983). Google Scholar

[25] 25.Ricceri, B., On three critical points theorem, Arch. Math. (Basel) 75 (2000), 220–226. Google Scholar | DOI

[26] 26.Ruzicka, M., Electrorheological fluids: Modelingn and mathematical theory (Springer-Verlag, Berlin, 2002). Google Scholar

[27] 27.Zhikov, V., Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. 29 (1987), 33–66. Google Scholar

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