Abstract
In this work, we consider second-order elliptic equations with nonlinearities determined by Musielak–Orlicz functions and summable right-hand side. The equivalence of entropy solutions and renormalized solutions of the Dirichlet problem in arbitrary unbounded domains with Lipschitz boundary is established. The existence and uniqueness of both entropy solutions and renormalized solutions are proved in nonreflexive Musielak–Orlicz–Sobolev spaces.
REFERENCES
P. Gwiazda, I. Skrzypczaka, and A. Zatorska-Goldstein, ‘‘Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space,’’ Differ. Equat. 264, 341–377 (2018).
A. Denkowska, P. Gwiazda, and P. Kalita, ‘‘On renormalized solutions to elliptic inclusions with nonstandard growth,’’ Calc. Var. Part. Differ. Equat. 60 (21), 1–44 (2021).
M. Ait Khellou and A. Benkirane, ‘‘Renormalized solution for nonlinear elliptic problems with lower order terms and \(L^{1}\) data in Musielak-Orlicz spaces,’’ Ann. Univ. Craiova, Math. Comput. Sci. 43, 164–187 (2016).
M. S. B. Elemine Vall, T. Ahmedatt, A. Touzani, and A. Benkirane, ‘‘Existence of entropy solutions for nonlinear elliptic equations in Musielak framework with \(L^{1}\) data,’’ Bol. Soc. Paran. Mat. 36, 125–150 (2018).
R. Elarabi, M. Rhoudaf, and H. Sabiki, ‘‘Entropy solution for a nonlinear elliptic problem with lower order term in Musielak–Orlicz spaces,’’ Ricerch. Mat. 67, 549–579 (2018).
M. Ait Khelloul, S. M. Douiri, and Y. El Hadfi, ‘‘Existence of solutions for some nonlinear elliptic equations in musielak spaces with only the Log-Hölder continuity condition,’’ Mediterr. J. Math. 17 (1), 1–18 (2020).
A. Talha and A. Benkirane, ‘‘Strongly nonlinear elliptic boundary value problems in Musielak–Orlicz spaces,’’ Monatsh. Math. 186, 745–776 (2018).
Li Ying, Y. Fengping, and Zh. Shulin, ‘‘Entropy and renormalized solutions to the general nonlinear elliptic equations in Musielak–Orlicz spaces,’’ Nonlin. Anal.: Real World Appl. 61, 1–20 (2021).
L. M. Kozhevnikova and A. P. Kashnikova, ‘‘Equivalence of entropy and renormalized solutions of a nonlinear elliptic problem in Musielak-Orlicz spaces,’’ Differ. Equat. 59 (1), 34–50 (2023).
M. A. Krasnosel’skii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces (P. Noordhoff, Groningen, 1961).
J. Musielak, Orlicz Spaces and Modular Spaces, Vol. 1034 of Lecture Notes in Mathematics (Springer, Berlin, 1983).
I. Chlebicka, ‘‘A pocket guide to nonlinear differential equations in Musielak-Orlicz spaces,’’ Nonlin. Anal. 175, 1–27 (2018).
Y. Ahmida, I. Chlebicka, P. Gwiazda, and A. Youssfi, ‘‘Gossez’s approximation theorems in Musielak–Orlicz–Sobolev spaces,’’ J. Funct. Anal. 275, 2538–2571 (2018).
N. Dunford and J. T. Schwartz, Linear Operators. Part 1: General Theory, Vol. 7 of Pure and Applied Mathematics (Interscience, New York, 1958).
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations (Academic, New York, 1968).
A. Benkirane and M. Sidi El Vally, ‘‘Variational inequalities in Musielak–Orlicz–Sobolev spaces,’’ Bull. Belg. Math. Soc. Simon Stevin. 21, 787–811 (2014).
L. Boccardo and Th. Gallouët, ‘‘Nonlinear elliptic equations with right-hand side measures,’’ Comm. Part. Differ. Equat. 17, 641–655 (1992).
I. Chlebicka, ‘‘Measure data elliptic problems with generalized Orlicz growth,’’ Proc. R. Soc. Edinburgh, Sect. A: Math., 1–31 (2022).
A. A. Kovalevsky, I. I. Skrypnik, and A. E. Shishkov, Singular Solutions of Nonlinear Elliptic and Parabolic Equations, Problems and Methods: Mathematics, Mechanics, Cybernetics (Naukova Dumka, Kyiv, 2010) [in Russian].
L. M. Kozhevnikova, ‘‘Entropy and renormalized solutions of anisotropic elliptic equations with variable nonlinearity exponents,’’ Sb. Math. 210, 417–446 (2019).
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by A. B. Muravnik)
Rights and permissions
About this article
Cite this article
Kozhevnikova, L.M. On Solutions of Nonlinear Elliptic Equations with \(\boldsymbol{L}_{\mathbf{1}}\)-Data in Unbounded Domains. Lobachevskii J Math 44, 1879–1901 (2023). https://doi.org/10.1134/S1995080223050372
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080223050372