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DOI: 10.46698/c8515-1572-8469-r
Existence and Uniqueness of Solution for Nonlinear Anisotropic Elliptic Dirichlet Problems
Naceri, M.
Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 1.
Abstract: We consider boundary value problem for nonlinear anisotropic elliptic partial differential equations in bounded open Lipschitz domain and the Dirichlet boundary conditions. We also suppose that the body force function belongs to the natural dual space under certain hypotheses regarding the nonlinear anisotropic operators present on the main side of the proposed problems. We prove the existence and uniqueness of a weak solution in anisotropic Sobolev space for this problem. Our proofs are based on various anisotropic Sobolev inequalities, embedding theorems, and features of pseudo-monotone operators. The functional setting involves anisotropic Lebesgue and Sobolev spaces in the scalar case and their most important properties.
For citation: Naceri, M. Existence and Uniqueness of Solution for Nonlinear Anisotropic Elliptic Dirichlet Problems, Vladikavkaz Math. J., 2026, vol. 28, no.1, pp.122-133. DOI 10.46698/c8515-1572-8469-r
1. Ruzicka, M. Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000.
2. Chen, Y., Levine, S. and Rao, M. Variable Exponent, Linear Growth Functionals in Image Restoration, SIAM Journal on Applied Mathematics, 2006, vol. 66, no. 4, pp. 1383-1406. DOI: 10.1137/050624522.
3. Bear, J. Dynamics of Fluids in Porous Media, New York, American Elsevier, 1972.
4. Bendahmane, M., Langlais, M. and Saad, M. On Some Anisotropic Reaction-Diffusion Systems with \(L^1\)-data Modeling the
Propagation of an Epidemic Disease, Nonlinear Analysis: Theory, Methods & Applications,
2003, vol. 54, no. 4, pp. 617-636. DOI: 10.1016/S0362-546X(03)00090-7.
5. Antontsev, S. and Chipot, M. Anisotropic Equations: Uniqueness and Existence Results, Differential Integral Equations, 2008, vol. 21, no. 5-6, pp. 401-419.
6. Boccardo, L., Gallouet, T. and Murat, F. Unicite de la Solution de Certaines Equations Elliptiques non Lineaires,
Comptes Rendus de l'Academie des Sciences. Serie I. Mathematique, 1992, vol. 315, no. 1, pp. 1159-1164.
7. Carrillo, J. and Chipot, M. On Some Nonlinear Elliptic Equations Involving Derivatives of the Nonlinearity, Proceedings of the Royal Society of Edinburgh, Section A Mathematics, 1985, vol. 100, no. 3-4, pp. 281-294.
8. Boccardo, L. and Cirmi, R. Existence and Uniqueness of Solutions of Unilateral Problems with \(L^1\) Data, Journal of Convex Analysis, 1999, vol. 6, no. 1, pp. 195-206.
9. Chipot, M. and Cimatti, G. A Uniqueness Result for the Thermistor Problem, European Journal of Applied Mathematics,
1991, vol. 2, no. 2, pp. 97-103.
10. Chipot, M. and Michaille, G. Uniqueness Results and Monotonicity Properties for Strongly Nonlinear Elliptic Variational Inequalities, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e serie, 1989,
vol. 16, no. 1, pp. 137-166.
11. Zeidler, E. Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators, Translated from the German by the Author and Leo F. Boron, New York, Springer-Verlag, 1990.
12. Brezis, H. Equations et Inequations Non Lineaires Dans les Espaces Vectoriels en Dualite,
Annales de l'Institut Fourier, 1968, vol. 18, no. 1, pp. 115-175.
13. Browder, F. E. Nonlinear Elliptic Boundary Value Problems, Bulletin of the American
Mathematical Society, 1963, vol. 69, no. 6, pp. 862-874. DOI: 10.1090/S0002-9904-1963-11068-X.
14. Minty, G. J. On a ``Monotonicity'' Method for the Solution of Nonlinear Equations in Banach Spaces, Proceedings of the National Academy of Sciences of the United States of America,
1963, vol. 50, no. 6, pp. 1038-1041. DOI: 10.1073/pnas.50.6.1038.
15. Nikolskii, S. M. Imbedding Theorems for Functions with Partial Derivatives Considered in Different Metrics, Doklady Akademii Nauk SSSR, 1958, vol. 118, no. 1, pp. 35-37 (in Russian).
16. Nikolskii, S. M. An Imbedding Theorem for Functions with Partial Derivatives Considered in Different Metrics, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 1958, vol. 22, no. 3, pp. 321-336 (in Russian).
17. Slobodeckii, L. N. Generalized Sobolev Spaces and Their Application to Boundary Problems for Partial Differential Equations, Leningradskij gosudarstvennyj pedagogicheskij institut: Uchenye zapiski [Leningrad State Pedagogical Institute: Scientific Notes], 1958, vol. 197, pp. 54-112 (in Russian).
18. Troisi, M. Teoremi di Inclusione per Spazi di Sobolev non Isotropi, Ricerche di Matematica,
1969, vol. 18, no. 3, pp. 3-24.
19. Trudinger, N. S. An Imbedding Theorem for \(H sb{0(G,\,\Omega)}\) Spaces, Studia Mathematica,
1974, vol. 50, no. 1, pp. 17-30.
20. Fragala, I. , Gazzola, F. and Kawohl, B. Existence and Nonexistence Results for Anisotropic Quasilinear Elliptic Equations, Annales de l'Institut Henri Poincare, 2004, vol. 21, no. 5, pp. 715-734.
21. Fan, X. Anisotropic Variable Exponent Sobolev Spaces and \(\overrightarrow{p(x)}\)-Laplacian Equations, Complex Variables and Elliptic Equations, 2011, vol. 56, no. 7-9, pp. 623-642. DOI: 10.1080/17476931003728412.
22. Naceri, M. Anisotropic Nonlinear Weighted Elliptic Equations with Variable Exponents, Georgian Mathematical Journal,
2023, vol. 30, no. 2, pp. 277-285. DOI: 10.1515/gmj-2022-2216.
23. Naceri, M. Entropy Solutions for Variable Exponents Nonlinear Anisotropic Elliptic Equations with Natural Growth Terms,
Revista Colombiana de Matematicas, 2024, vol. 58, no. 1, pp. 99-115. DOI: 10.15446/recolma.v58n1.117442.
24. Naceri, M. Anisotropic Nonlinear Elliptic Systems with Variable Exponents, Degenerate Coercivity and \(L^{q(\cdot)}\) Data, Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications,
2022, vol. 14, no. 1-2, pp. 107-140. DOI: 10.56082/annalsarscimath.2022.1-2.107.
25. Naceri, M. and Benboubker, M. B. Distributional Solutions of Anisotropic Nonlinear Elliptic Systems with Variable Exponents: Existence and Regularity, Advances in Operator Theory,
2022, vol. 7, no. 2, pp. 1-34. DOI: 10.1007/s43036-022-00183-4.
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