Abstract
We study in Musielak-Sobolev space, the existence of entropy solution for a class of nonlinear elliptic problems, in the case the Leray-Lions operator satisfies only the large monotonicity condition and right hand side \(f\in W^{-1} L_{\psi }(\Omega )\).
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Ait Khellou, M., Benkirane, A., Douiri, S.M.: An inequality of type Poincaré in Musielak spaces and application to some non-linear elliptic problems with data. Complex Variables and Elliptic Equations. 60(9), 1217–1242 (2015)
Aissaoui, F., Benkirane, A., El Moumni, M., Youssfi, A.: Existence of renormalized solutions for some strongly nonlinear elliptic equations in Orlicz spaces. Georgian Math. J. 22(3), 305–321 (2015)
Benkirane, A., El Moumni, M., Kouhaila, K.: Solvability of strongly nonlinear elliptic variational problems in weighted OrliczSobolev spaces. SeMA Journal 77, 119–142 (2020)
Bénilan, Ph., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.L.: An \(L^1-\)theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann Scuola Norm Sup PisaCl Sci. 4(2), 241–273 (1995)
Boccardo, L.: Positive solutions for some quasilinear elliptic equations with natural growths. Atti Accad. Naz. Lincei 11(1), 31–39 (2000)
Boccardo, L., Gallouët, T.: Nonlinear elliptic equations with right-hand side measures. Commun. Partial Differ. Equ. 17, 641–655 (1992)
Boccardo, L., Gallouët, T., Vázquez, J.L.: Nonlinear elliptic equations in \({\mathbb{R} }^N\) without growth restrictions on the data. J. Differential Equations 105(2), 334–363 (1993)
Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)
Brezis, H.: Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies, No. 5. Notas de Matemàtica (50). North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973, MR 50 \(\ne \) 1060 Zbl 252.47055
Browder, F. E.: Existence theorems for nonlinear partial differential equations, Global Analysis (Berkeley, 1968), Proc. Sympos. Pure Math., no. XVI, AMS, Providence, 1970, pp. 1–60, MR 42 6= 4855
Cheng, C., Xu, J.: Geometric properties of Banach space valued Bochner-Lebesgue spaces with variable exponent. J. Math. Inequal. 7, 461–475 (2013)
Chen, Y., Levine, S., Rao, R.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)
Diening, L., Harjulehto, P., Hasto, P., Ruzuka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Berlin (2011)
El Amarty, N., El Haji, B., El Moumni, M.: Existence of renomalized solution for nonlinear elliptic boundary value problem without \(\Delta _2 \)-condition. SeMA Journal 77(4), 389–414 (2020)
El Haji, B., El Moumni, M., Kouhaila, K.: On a nonlinear elliptic problems having large monotonocity with \(L^1\)-data in weighted Orlicz-Sobolev spaces. Moroccan J. of Pure and Appl. Anal. (MJPAA). 5(1), 104–116 (2019)
El Moumni, M.: Nonlinear elliptic equations without sign condition and \(L^1\)-data in Musielak-Orlicz-Sobolev spaces. Acta. Appl. Math. 59, 95–117 (2019)
El Moumni, M.: Renormalized solutions for strongly nonlinear elliptic problems with lower order terms and measure data in Orlicz-Sobolev spaces. Iran. J. Math. Sci. Inform. 14(1), 95–119 (2019)
El Moumni, M.: Entropy solution for strongly nonlinear elliptic problems with lower order terms and \(L^1\)-data. Annals of the University of Craiova - Mathematics and Computer Science Series. 40(2), 211–225 (2013)
Leray, J., Lions, J.L.: Quelques rêsultats de visik sur les problèmes elliptiques semilinèaires par les mèthodes de Minty et Browder. Bull. Soc. Math. France 93, 97–107 (1965)
Li, F., Li, Z., Pi, L.: Variable exponent functionals in image restoration. Appl. Math. Comput. 216, 870–882 (2010)
Lions, J.L.: Quelques mèthodes de rèsolution des problèmes aux limites non linèaire. Dunod et Gauthier Villars, Paris (1969)
Minty, G.J.: On a monotonicity method for the solution of non-linear equations in Banach spaces. Proc. Nat. Acad. Sci. U.S.A. 50, 1038–1041 (1963)
Minty, G.J.: Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 1962
Musielak, J.: Orlicz spaces and modular spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)
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El Amarty, N., El Moumni, M. & Bouzyani, R. Variational nonlinear elliptic equations having large monotonocity in Musielak-Sobolev spaces. SeMA (2024). https://doi.org/10.1007/s40324-024-00349-5
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DOI: https://doi.org/10.1007/s40324-024-00349-5