Abstract
In this paper we use the penalization method to prove the existence of solution for variational inequalities of Leray–Lions type, in the setting of Musielak spaces and where the Musielak function doesn’t satisfy the \(\Delta _2\)-condition. Here the right-hand side is in \(L^1.\)
Similar content being viewed by others
References
Aharouch, L., Azroul, E., Rhoudaf, M.: Existence of solutions for unilateral problems in \(L^1\) involving lower order terms in divergence form in Orlicz spaces. J. Appl. Anal. 13, 151–181 (2007)
Ait Khellou, M., Benkirane, A., Douiri, S.M.: Existence of solutions for elliptic equations having natural growth terms in Musielak–Orlicz spaces. J. Math. Comput. Sci. 4(4), 665–688 (2014)
Ait Khellou, M., Benkirane, A., Douiri, S.M.: Some properties of Musielak spaces with only the log-Hölder continuity condition and application. Ann. Funct. Anal. 11, 1062–1080 (2020)
Benkirane, A., Elmahi, A., Meskine, D.: On the limit of some penalized problems involving increasing powers. Asymptot. Anal. 36, 303–317 (2003)
Benkirane, A., Sidi El Vally, M.: Variational inequalities in Musielak–Orlicz–Sobolev spaces. Bull. Belg. Math. Soc. Simon Stevin 21, 787–811 (2014)
Benkirane, A., Sidi El Vally, M.S.: Variational inequalities in Musielak–Orlicz–Sobolev spaces. Bull. Belg. Math. Soc. Simon Stevin 21(5), 787–811 (2014)
Benkirane, A., Sidi El Vally, M.S.: Some approximation properties in Musielak–Orlicz–Sobolev spaces. Thai J. Math. 10(2), 371–381 (2012)
Dall’aglio, A., Orsina, L.: On the limit of some nonlinear elliptic equations involving increasing powers. Asympt. Anal. 14, 49–71 (1997)
Elarabi, R., Rhoudaf, M., Sabiki, H.: Entropy solution for a nonlinear elliptic problem with lower order term in Musielak–Orlicz spaces. Ric. Mat., 1–31 (2017)
Gossez, J.-P.: Some approximation properties in Orlicz–Sobolev. Studia Math. 74, 17–24 (1982)
Musielak, J.: Modular Spaces and Orlicz Spaces. Lecture Notes in Mathematics, vol. 1034. Springer Verlag, Berlin (1983)
Rajagopal, K.R., Ru̇z̃ic̃ka, M.: Mathematical modeling of electrorheological materials. Contin. Mech. Thermodyn. 13, 59–78 (2001)
Ru̇žic̆ka, M.: Electrorheological Fluids, Modeling and Mathematical Theory. Lecture Notes in Mathematics. Springer, Berlin (2000)
Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Elarabi, R., Rhoudaf, M. Some obstacle problems in Musielak spaces. Ricerche mat (2022). https://doi.org/10.1007/s11587-021-00679-w
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11587-021-00679-w