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]

Mihăilescu, Mihai ; Rădulescu, Vicenţiu

Annales de l'Institut Fourier, Tome 58 (2008), p. 2087-2111 / Harvested from Numdam

Résumé

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.

Abstract

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.

Détail

Publié le : 2008-01-01

MR 2473630 | Zbl pre05367570 | Zbl 1186.35065

DOI : https://doi.org/10.5802/aif.2407
Classification:  35D05,  35J60,  35J70,  58E05,  68T40,  76A02
Mots clés: opérateur différentiel non homogène, équation aux dérivées partielles non linéaire, problème de Neumann, espace d’Orlicz–Sobolev

Bibliographie

[1] Acerbi, Emilio; Mingione, Giuseppe Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., Tome 156 (2001) no. 2, pp. 121-140 | Article | MR 1814973 | Zbl 0984.49020

[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 450957 | Zbl 0314.46030

[3] Brezis, Haïm Analyse fonctionnelle, Masson, Paris, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree] (1983) (Théorie et applications. [Theory and applications]) | Zbl 0511.46001

[4] Chen, Yunmei; Levine, Stacey; Rao, Murali Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., Tome 66 (2006) no. 4, p. 1383-1406 (electronic) | Article | MR 2246061 | Zbl 1102.49010

[5] Clément, Ph.; García-Huidobro, M.; Manásevich, R.; Schmitt, K. Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, Tome 11 (2000) no. 1, pp. 33-62 | Article | MR 1777463 | Zbl 0959.35057

[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., Tome 1 (2004) no. 3, pp. 241-267 | Article | MR 2094464 | Zbl pre02216926

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

[8] Diening, Lars Theorical and numerical results for electrorheological fluids, University of Freiburg (2002) (Ph. D. Thesis) | Zbl 1022.76001

[9] Diening, Lars Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., Tome 129 (2005) no. 8, pp. 657-700 | Article | MR 2166733 | Zbl 1096.46013

[10] Donaldson, Thomas K.; Trudinger, Neil S. Orlicz-Sobolev spaces and imbedding theorems, J. Functional Analysis, Tome 8 (1971), pp. 52-75 | Article | MR 301500 | Zbl 0216.15702

[11] Edmunds, D. E.; Lang, J.; Nekvinda, A. On $L^{p(x)}$ norms, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., Tome 455 (1999) no. 1981, pp. 219-225 | Article | MR 1700499 | Zbl 0953.46018

[12] Edmunds, David E.; Rákosník, Jiří Density of smooth functions in $W^{k,p(x)}(\Omega )$, Proc. Roy. Soc. London Ser. A, Tome 437 (1992) no. 1899, pp. 229-236 | Article | MR 1177754 | Zbl 0779.46027

[13] Edmunds, David E.; Rákosník, Jiří Sobolev embeddings with variable exponent, Studia Math., Tome 143 (2000) no. 3, pp. 267-293 | MR 1815935 | Zbl 0974.46040

[14] Ekeland, I. On the variational principle, J. Math. Anal. Appl., Tome 47 (1974), pp. 324-353 | Article | MR 346619 | Zbl 0286.49015

[15] Fan, Xianling; Shen, Jishen; Zhao, Dun Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega )$, J. Math. Anal. Appl., Tome 262 (2001) no. 2, pp. 749-760 | Article | MR 1859337 | Zbl 0995.46023

[16] Fan, Xianling; Zhao, Dun On the spaces $L^{p(x)}(\Omega )$ and $W^{m,p(x)}(\Omega )$, J. Math. Anal. Appl., Tome 263 (2001) no. 2, pp. 424-446 | Article | MR 1866056 | Zbl 1028.46041

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

[18] Kováčik, Ondrej; Rákosník, Jiří On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., Tome 41(116) (1991) no. 4, pp. 592-618 | MR 1134951 | Zbl 0784.46029

[19] Lamperti, John On the isometries of certain function-spaces, Pacific J. Math., Tome 8 (1958), pp. 459-466 | MR 105017 | Zbl 0085.09702

[20] Marcellini, Paolo Regularity and existence of solutions of elliptic equations with $p,q$-growth conditions, J. Differential Equations, Tome 90 (1991) no. 1, pp. 1-30 | Article | MR 1094446 | Zbl 0724.35043

[21] Mihăilescu, Mihai; Pucci, Patrizia; Rădulescu, Vicenţiu Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C. R. Math. Acad. Sci. Paris, Tome 345 (2007) no. 10, pp. 561-566 | MR 2374465 | Zbl 1127.35020

[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., Tome 462 (2006) no. 2073, pp. 2625-2641 | Article | MR 2253555 | Zbl pre05278124

[23] Mihăilescu, Mihai; Rădulescu, Vicenţiu Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl., Tome 330 (2007) no. 1, pp. 416-432 | Article | MR 2302933 | Zbl pre05141592

[24] Mihăilescu, Mihai; Rădulescu, Vicenţiu On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., Tome 135 (2007) no. 9, p. 2929-2937 (electronic) | Article | MR 2317971 | Zbl pre05165473

[25] Mihăilescu, Mihai; Rădulescu, Vicenţiu Continuous spectrum for a class of nonhomogeneous differential operators, Manuscripta Math., Tome 125 (2008) no. 2, pp. 157-167 | Article | MR 2373080 | Zbl 1138.35070

[26] Musielak, J.; Orlicz, W. On modular spaces, Studia Math., Tome 18 (1959), pp. 49-65 | MR 101487 | Zbl 0086.08901

[27] Musielak, Julian Orlicz spaces and modular spaces, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1034 (1983) | MR 724434 | Zbl 0557.46020

[28] Nakano, Hidegorô Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo (1950) | MR 38565 | Zbl 0041.23401

[29] O’Neill, R. Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc., Tome 115 (1965), pp. 300-328 | Article | Zbl 0132.09201

[30] Orlicz, Władysław Über konjugierte Exponentenfolgen, Studia Math., Tome 3 (1931), pp. 200-211 | Zbl 0003.25203

[31] Rajagopal, K. R.; Růžička, M. Mathematical modelling of electrorheological fluids, Cont. Mech. Term., Tome 13 (2001), pp. 59-78 | Article | Zbl 0971.76100

[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 1788852 | Zbl 0968.76531

[33] Struwe, Michael Variational methods, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 34 (1996) (Applications to nonlinear partial differential equations and Hamiltonian systems) | MR 1411681 | Zbl 0864.49001

[34] Zhikov, V. V. Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., Tome 50 (1986) no. 4, p. 675-710, 877 | MR 864171 | Zbl 0599.49031

[35] Zhikov, V. V. Meyer-type estimates for solving the nonlinear Stokes system, Differ. Uravn., Tome 33 (1997) no. 1, p. 107-114, 143 | MR 1607245 | Zbl 0911.35089