Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces
[Problèmes de Neumann associés aux opérateurs différentiels non homogènes dans les espaces d’Orlicz–Sobolev]
Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2087-2111.

On étudie un problème aux limites de Neumann associé à un opérateur différentiel non homogène. En tenant compte de la compétition entre le taux de croissance de la nonlinéarité et les valeurs du paramètre de bifurcation, on établit des conditions suffisantes pour l’existence de solutions non triviales dans un certain espace fonctionnel du type Orlicz–Sobolev.

We study a nonlinear Neumann boundary value problem associated to a nonhomogeneous differential operator. Taking into account the competition between the nonlinearity and the bifurcation parameter, we establish sufficient conditions for the existence of nontrivial solutions in a related Orlicz–Sobolev space.

DOI : 10.5802/aif.2407
Classification : 35D05, 35J60, 35J70, 58E05, 68T40, 76A02
Keywords: Nonhomogeneous differential operator, nonlinear partial differential equation, Neumann boundary value problem, Orlicz–Sobolev space
Mot clés : opérateur différentiel non homogène, équation aux dérivées partielles non linéaire, problème de Neumann, espace d’Orlicz–Sobolev
Mihăilescu, Mihai 1 ; Rădulescu, Vicenţiu 2

1 University of Craiova Department of Mathematics 200585 Craiova (Romania) Central European University Department of Mathematics 1051 Budapest (Hungary)
2 Institute of Mathematics “Simion Stoilow” of the Romanian Academy P.O. Box 1-764 014700 Bucharest (Romania) University of Craiova Department of Mathematics 200585 Craiova (Romania)
@article{AIF_2008__58_6_2087_0,
     author = {Mih\u{a}ilescu, Mihai and R\u{a}dulescu, Vicen\c{t}iu},
     title = {Neumann problems associated to nonhomogeneous differential operators in {Orlicz{\textendash}Sobolev} spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {2087--2111},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {6},
     year = {2008},
     doi = {10.5802/aif.2407},
     mrnumber = {2473630},
     zbl = {1186.35065},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2407/}
}
TY  - JOUR
AU  - Mihăilescu, Mihai
AU  - Rădulescu, Vicenţiu
TI  - Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces
JO  - Annales de l'Institut Fourier
PY  - 2008
SP  - 2087
EP  - 2111
VL  - 58
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2407/
DO  - 10.5802/aif.2407
LA  - en
ID  - AIF_2008__58_6_2087_0
ER  - 
%0 Journal Article
%A Mihăilescu, Mihai
%A Rădulescu, Vicenţiu
%T Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces
%J Annales de l'Institut Fourier
%D 2008
%P 2087-2111
%V 58
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2407/
%R 10.5802/aif.2407
%G en
%F AIF_2008__58_6_2087_0
Mihăilescu, Mihai; Rădulescu, Vicenţiu. Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces. Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2087-2111. doi : 10.5802/aif.2407. http://www.numdam.org/articles/10.5802/aif.2407/

[1] Acerbi, Emilio; Mingione, Giuseppe Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., Volume 156 (2001) no. 2, pp. 121-140 | DOI | MR | Zbl

[2] Adams, Robert A. Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975 (Pure and Applied Mathematics, Vol. 65) | MR | Zbl

[3] Brezis, Haïm Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (Théorie et applications. [Theory and applications]) | Zbl

[4] Chen, Yunmei; Levine, Stacey; Rao, Murali Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., Volume 66 (2006) no. 4, p. 1383-1406 (electronic) | DOI | MR | Zbl

[5] Clément, Ph.; García-Huidobro, M.; Manásevich, R.; Schmitt, K. Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, Volume 11 (2000) no. 1, pp. 33-62 | DOI | MR | Zbl

[6] Clément, Philippe; de Pagter, Ben; Sweers, Guido; de Thélin, François Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math., Volume 1 (2004) no. 3, pp. 241-267 | DOI | MR

[7] Dankert, Gabriele Sobolev Embedding Theorems in Orlicz Spaces, University of Köln (1966) (Ph. D. Thesis Ph. D. Thesis)

[8] Diening, Lars Theorical and numerical results for electrorheological fluids, University of Freiburg (2002) (Ph. D. Thesis Ph. D. Thesis) | Zbl

[9] Diening, Lars Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., Volume 129 (2005) no. 8, pp. 657-700 | DOI | MR | Zbl

[10] Donaldson, Thomas K.; Trudinger, Neil S. Orlicz-Sobolev spaces and imbedding theorems, J. Functional Analysis, Volume 8 (1971), pp. 52-75 | DOI | MR | Zbl

[11] Edmunds, D. E.; Lang, J.; Nekvinda, A. On L p(x) norms, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., Volume 455 (1999) no. 1981, pp. 219-225 | DOI | MR | Zbl

[12] Edmunds, David E.; Rákosník, Jiří Density of smooth functions in W k,p(x) (Ω), Proc. Roy. Soc. London Ser. A, Volume 437 (1992) no. 1899, pp. 229-236 | DOI | MR | Zbl

[13] Edmunds, David E.; Rákosník, Jiří Sobolev embeddings with variable exponent, Studia Math., Volume 143 (2000) no. 3, pp. 267-293 | MR | Zbl

[14] Ekeland, I. On the variational principle, J. Math. Anal. Appl., Volume 47 (1974), pp. 324-353 | DOI | MR | Zbl

[15] Fan, Xianling; Shen, Jishen; Zhao, Dun Sobolev embedding theorems for spaces W k,p(x) (Ω), J. Math. Anal. Appl., Volume 262 (2001) no. 2, pp. 749-760 | DOI | MR | Zbl

[16] Fan, Xianling; Zhao, Dun On the spaces L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl., Volume 263 (2001) no. 2, pp. 424-446 | DOI | MR | Zbl

[17] Halsey, Thomas C. Electrorheological Fluids, Science, Volume 258 (1992) no. 5083, pp. 761 -766 | DOI

[18] Kováčik, Ondrej; Rákosník, Jiří On spaces L p(x) and W k,p(x) , Czechoslovak Math. J., Volume 41(116) (1991) no. 4, pp. 592-618 | MR | Zbl

[19] Lamperti, John On the isometries of certain function-spaces, Pacific J. Math., Volume 8 (1958), pp. 459-466 | MR | Zbl

[20] Marcellini, Paolo Regularity and existence of solutions of elliptic equations with p,q-growth conditions, J. Differential Equations, Volume 90 (1991) no. 1, pp. 1-30 | DOI | MR | Zbl

[21] Mihăilescu, Mihai; Pucci, Patrizia; Rădulescu, Vicenţiu Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C. R. Math. Acad. Sci. Paris, Volume 345 (2007) no. 10, pp. 561-566 | MR | Zbl

[22] Mihăilescu, Mihai; Rădulescu, Vicenţiu A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 462 (2006) no. 2073, pp. 2625-2641 | DOI | MR

[23] Mihăilescu, Mihai; Rădulescu, Vicenţiu Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl., Volume 330 (2007) no. 1, pp. 416-432 | DOI | MR

[24] Mihăilescu, Mihai; Rădulescu, Vicenţiu On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., Volume 135 (2007) no. 9, p. 2929-2937 (electronic) | DOI | MR

[25] Mihăilescu, Mihai; Rădulescu, Vicenţiu Continuous spectrum for a class of nonhomogeneous differential operators, Manuscripta Math., Volume 125 (2008) no. 2, pp. 157-167 | DOI | MR | Zbl

[26] Musielak, J.; Orlicz, W. On modular spaces, Studia Math., Volume 18 (1959), pp. 49-65 | MR | Zbl

[27] Musielak, Julian Orlicz spaces and modular spaces, Lecture Notes in Mathematics, 1034, Springer-Verlag, Berlin, 1983 | MR | Zbl

[28] Nakano, Hidegorô Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo, 1950 | MR | Zbl

[29] O’Neill, R. Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc., Volume 115 (1965), pp. 300-328 | DOI | Zbl

[30] Orlicz, Władysław Über konjugierte Exponentenfolgen, Studia Math., Volume 3 (1931), pp. 200-211 | Zbl

[31] Rajagopal, K. R.; Růžička, M. Mathematical modelling of electrorheological fluids, Cont. Mech. Term., Volume 13 (2001), pp. 59-78 | DOI | Zbl

[32] Růžička, M. Electrorheological fluids: modeling and mathematical theory, Sūrikaisekikenkyūsho Kōkyūroku (2000) no. 1146, pp. 16-38 Mathematical analysis of liquids and gases (Japanese) (Kyoto, 1999) | MR | Zbl

[33] Struwe, Michael Variational methods, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 34, Springer-Verlag, Berlin, 1996 (Applications to nonlinear partial differential equations and Hamiltonian systems) | MR | Zbl

[34] Zhikov, V. V. Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., Volume 50 (1986) no. 4, p. 675-710, 877 | MR | Zbl

[35] Zhikov, V. V. Meyer-type estimates for solving the nonlinear Stokes system, Differ. Uravn., Volume 33 (1997) no. 1, p. 107-114, 143 | MR | Zbl

Cité par Sources :