The Entropy Solutions for Anisotropic Neumann Problems with Variable Exponents and Hardy Potential
Entropy Solutions for Anisotropic Neumann Problems
Abstract
In this work, we study an anisotropic obstacle problem driven by a Leray-Lions-type operator with a Hardy-type singular potential,defined in anisotropic weighted Sobolev spaces with variable exponents. The problem involves a nonlinear lower-order term depending on
the gradient, under homogeneous Neumann boundary conditions. We establish the existence of entropy solutions using truncation techniques
combined with the monotonicity method.
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References
\begin{thebibliography}{99}
\bibitem{kass_1}Abbassi, A., Allalou, C., Kassidi, A.: Anisotropic Elliptic Nonlinear Obstacle Problem with Weighted Variable Exponent. J. Math. Study, \textbf{54}(4),(2021)337-356.
\bibitem{Antontsev_1} Antontsev, S., Chipot, M., Anisotropic equations: uniqueness and existence results. Differential Integral Equations 21(5-6), 401-419 (2008).
\bibitem{hjiej3}Azroul, E., Benboubker, M. B., Hjiaj, H., Yazough, C.: Existence of solutions for a class of obstacle problems with $ L^1$-data and without sign condition. Afrika Matematika, \textbf{27}(5-6), 795-813 (2016).
\bibitem{Azroul_1}Azroul, E., Bouziani, M., Barbara, A.: Existence of entropy solutions for anisotropic quasilinear degenerated elliptic problems with Hardy potential. SeMA Journal, \textbf{78}, 475-499(2021).
\bibitem{BenkiraneElmahi}Benkirane, A., Elmahi, A.: Strongly nonlinear elliptic unilateral problems having natural growth terms and $L^{1}$ data. Rendiconti di matematica, Serie VII, \textbf{18}, 289-303(1998).
\bibitem{Boccardo}Boccardo, L., Murat, F., Puel, J.P.: Existence of bounded solutions for non linear elliptic unilateral problems. Ann. Mat. Pura Appl, \textbf{(4)}152, 183-196(1988)
%
\bibitem{brezisStauss}Brezis, H. Strauss W: Semilinear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, \textbf{25}(4), 565-590(1973).
\bibitem{Brezis1992}Brezis, H.: Analyse Fonctionnelle. Th\'{e}orie, M\'{e}thodes et Applications. Masson, Paris (1992)
\bibitem{Chen}Chen, Y., Levine, S., Rao, R.: Rao. Functionals with $p(x)$-growth in image processing. Preprint (2004).
\bibitem{Chrif}Chrif, M., El Manouni, S.: Anisotropic equations in weighted Sobolev spaces of higher order. Ricerche di matematica, \textbf{58(1)}, 1-14 (2009).
\bibitem{Edmunds}Edmunds, D. E., Lang, J., Nekvinda, A.: On $L^{p(x)}$ norms. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, \textbf{455.1981}, 219-225(1999).
\bibitem{Edmunds_1}Edmunds, D. E., R\'{i}kosn\'{i}k, J.: Density of smooth functions in $W^{k, p(x)}(\Omega)$. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences \textbf{437.1899} : 229-236(1992).
\bibitem{fz}Fan, X. L., Zhang, Q. H.: Existence for $p(x)$-Laplacien Direchlet problem, Nonlinear Analysis, \textbf{52}, 1843-1852(2003)
\bibitem{Halsey}Halsey, T.C.: Electrorheological fluids, Science \textbf{258}, pp. 761-766(1992).
\bibitem{Kim}Kim, Y.-H., Wang, L., Zhang, C.: Global bifurcation for a class of degenerate elliptic equations with variable exponents. Journal of Mathematical Analysis and Applications, \textbf{371}, 624-637(2010).
\bibitem{kovacik}Kovacik, O., Rakosnik J.: On spaces $L^{p(x)}$ and $W^{1,p(x)}$, Czechoslovak Math. J. \textbf{41}, 592-618(1991).
\bibitem{Leone}Leone, C., Porretta, A. : Entropy solutions for nonlinear elliptic equations in $L^1$, Nonlinear Anal., \textbf{32}, 325-334 (1998).
\bibitem{Niko}Nikol'skii, S.M.: On imbedding, continuation and approximation theorems for differentiable functions of several variables. Russian Mathematical Surveys, \textbf{16.5}, 55(1961).
\bibitem{Rakosnik}R\~{a}kosn\'{i}k, J.: Some remarks to anisotropic Sobolev spaces I. Beitr\"{a}ge zur Analysis, \textbf{13.13-14}, 55-88(1979).
\bibitem{Ruzicka}Ruzicka, M.: Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, (2002).
\bibitem{sabiry2025analysis} Sabiry, A., Zineddaine, G., Chadli, L. S., \& Kassidi, A.: Analysis of Capillary Effects Modelled by Elliptic Equations with $\varrho(x)$-Laplacian-Like Operators and Hardy Potential. International Journal of Applied and Computational Mathematics, 11(1), 1-22(2025).
\bibitem{sabiry2024double}Sabiry, A., Zineddaine, G., Melliani, S., \& Kassidi, A.: The double obstacle problem in generalized Sobolev spaces with Hardy potential. Complex Variables and Elliptic Equations, 1-10(2024).
\bibitem{Salmani}Salmani, A., Akdim, Y., Redwane, H. : Entropy solutions of anisotropic elliptic nonlinear obstacle problem with measure data. Ricerche di Matematica, 1-31(2019).
\bibitem{Winslow}Winslow, W.: Method and means for translating electrical impulses into mechanical force, U.S. Patent \textbf{2.417.850}(1947).
\bibitem{Winslow_1}Winslow, W.: Induced fibration of suspensions, J. Appl. Phys. \textbf{20}, pp. 1137-1140(1949).
\bibitem{Youssfi}Youssfi, A., Azroul, E., Hjiaj, H.: On nonlinear elliptic equations with Hardy potentiel and $L^1$ data. Monatsh Math. \textbf{173}(1), 107-129 (2014)
\bibitem{zq}Zhao, D., Qiang, X. J., Fan, X. L.: On generalized Orlicz Spaces $L^{p(x)}(\Omega)$, J. Gansu Sci, \textbf{9}(2), 1-7(1997)
\bibitem{Zhikov}Zhikov V.V: Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. \textbf{29}, 33-66(1987).
\bibitem{zineddaine2024anisotropic} Zineddaine, G., Sabiry, A., Melliani, S., \& Kassidi, A.: Anisotropic obstacle Neumann problems in weighted Sobolev spaces and variable exponent. Journal of Applied Analysis, (0)(2024).
\bibitem{zineddaine2024ani} Zineddaine, G., Sabiry, A., Melliani, S., \& Kassidi, A.: Anisotropic obstacle Neumann problems in weighted Sobolev spaces with Hardy potential and variable exponent. SeMA Journal, 1-24(2024).
\bibitem{zineddaine2024existence}Zineddaine, G., Sabiry, A., Melliani, S., \& Kassidi, A. . Existence results in weighted Sobolev space for quasilinear degenerate $p(z)$-elliptic problems with a Hardy potential. Mathematical Modelling and Analysis, 29(3), 460-479(2024).
\end{thebibliography}
\bibitem{kass_1}Abbassi, A., Allalou, C., Kassidi, A.: Anisotropic Elliptic Nonlinear Obstacle Problem with Weighted Variable Exponent. J. Math. Study, \textbf{54}(4),(2021)337-356.
\bibitem{Antontsev_1} Antontsev, S., Chipot, M., Anisotropic equations: uniqueness and existence results. Differential Integral Equations 21(5-6), 401-419 (2008).
\bibitem{hjiej3}Azroul, E., Benboubker, M. B., Hjiaj, H., Yazough, C.: Existence of solutions for a class of obstacle problems with $ L^1$-data and without sign condition. Afrika Matematika, \textbf{27}(5-6), 795-813 (2016).
\bibitem{Azroul_1}Azroul, E., Bouziani, M., Barbara, A.: Existence of entropy solutions for anisotropic quasilinear degenerated elliptic problems with Hardy potential. SeMA Journal, \textbf{78}, 475-499(2021).
\bibitem{BenkiraneElmahi}Benkirane, A., Elmahi, A.: Strongly nonlinear elliptic unilateral problems having natural growth terms and $L^{1}$ data. Rendiconti di matematica, Serie VII, \textbf{18}, 289-303(1998).
\bibitem{Boccardo}Boccardo, L., Murat, F., Puel, J.P.: Existence of bounded solutions for non linear elliptic unilateral problems. Ann. Mat. Pura Appl, \textbf{(4)}152, 183-196(1988)
%
\bibitem{brezisStauss}Brezis, H. Strauss W: Semilinear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, \textbf{25}(4), 565-590(1973).
\bibitem{Brezis1992}Brezis, H.: Analyse Fonctionnelle. Th\'{e}orie, M\'{e}thodes et Applications. Masson, Paris (1992)
\bibitem{Chen}Chen, Y., Levine, S., Rao, R.: Rao. Functionals with $p(x)$-growth in image processing. Preprint (2004).
\bibitem{Chrif}Chrif, M., El Manouni, S.: Anisotropic equations in weighted Sobolev spaces of higher order. Ricerche di matematica, \textbf{58(1)}, 1-14 (2009).
\bibitem{Edmunds}Edmunds, D. E., Lang, J., Nekvinda, A.: On $L^{p(x)}$ norms. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, \textbf{455.1981}, 219-225(1999).
\bibitem{Edmunds_1}Edmunds, D. E., R\'{i}kosn\'{i}k, J.: Density of smooth functions in $W^{k, p(x)}(\Omega)$. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences \textbf{437.1899} : 229-236(1992).
\bibitem{fz}Fan, X. L., Zhang, Q. H.: Existence for $p(x)$-Laplacien Direchlet problem, Nonlinear Analysis, \textbf{52}, 1843-1852(2003)
\bibitem{Halsey}Halsey, T.C.: Electrorheological fluids, Science \textbf{258}, pp. 761-766(1992).
\bibitem{Kim}Kim, Y.-H., Wang, L., Zhang, C.: Global bifurcation for a class of degenerate elliptic equations with variable exponents. Journal of Mathematical Analysis and Applications, \textbf{371}, 624-637(2010).
\bibitem{kovacik}Kovacik, O., Rakosnik J.: On spaces $L^{p(x)}$ and $W^{1,p(x)}$, Czechoslovak Math. J. \textbf{41}, 592-618(1991).
\bibitem{Leone}Leone, C., Porretta, A. : Entropy solutions for nonlinear elliptic equations in $L^1$, Nonlinear Anal., \textbf{32}, 325-334 (1998).
\bibitem{Niko}Nikol'skii, S.M.: On imbedding, continuation and approximation theorems for differentiable functions of several variables. Russian Mathematical Surveys, \textbf{16.5}, 55(1961).
\bibitem{Rakosnik}R\~{a}kosn\'{i}k, J.: Some remarks to anisotropic Sobolev spaces I. Beitr\"{a}ge zur Analysis, \textbf{13.13-14}, 55-88(1979).
\bibitem{Ruzicka}Ruzicka, M.: Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, (2002).
\bibitem{sabiry2025analysis} Sabiry, A., Zineddaine, G., Chadli, L. S., \& Kassidi, A.: Analysis of Capillary Effects Modelled by Elliptic Equations with $\varrho(x)$-Laplacian-Like Operators and Hardy Potential. International Journal of Applied and Computational Mathematics, 11(1), 1-22(2025).
\bibitem{sabiry2024double}Sabiry, A., Zineddaine, G., Melliani, S., \& Kassidi, A.: The double obstacle problem in generalized Sobolev spaces with Hardy potential. Complex Variables and Elliptic Equations, 1-10(2024).
\bibitem{Salmani}Salmani, A., Akdim, Y., Redwane, H. : Entropy solutions of anisotropic elliptic nonlinear obstacle problem with measure data. Ricerche di Matematica, 1-31(2019).
\bibitem{Winslow}Winslow, W.: Method and means for translating electrical impulses into mechanical force, U.S. Patent \textbf{2.417.850}(1947).
\bibitem{Winslow_1}Winslow, W.: Induced fibration of suspensions, J. Appl. Phys. \textbf{20}, pp. 1137-1140(1949).
\bibitem{Youssfi}Youssfi, A., Azroul, E., Hjiaj, H.: On nonlinear elliptic equations with Hardy potentiel and $L^1$ data. Monatsh Math. \textbf{173}(1), 107-129 (2014)
\bibitem{zq}Zhao, D., Qiang, X. J., Fan, X. L.: On generalized Orlicz Spaces $L^{p(x)}(\Omega)$, J. Gansu Sci, \textbf{9}(2), 1-7(1997)
\bibitem{Zhikov}Zhikov V.V: Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. \textbf{29}, 33-66(1987).
\bibitem{zineddaine2024anisotropic} Zineddaine, G., Sabiry, A., Melliani, S., \& Kassidi, A.: Anisotropic obstacle Neumann problems in weighted Sobolev spaces and variable exponent. Journal of Applied Analysis, (0)(2024).
\bibitem{zineddaine2024ani} Zineddaine, G., Sabiry, A., Melliani, S., \& Kassidi, A.: Anisotropic obstacle Neumann problems in weighted Sobolev spaces with Hardy potential and variable exponent. SeMA Journal, 1-24(2024).
\bibitem{zineddaine2024existence}Zineddaine, G., Sabiry, A., Melliani, S., \& Kassidi, A. . Existence results in weighted Sobolev space for quasilinear degenerate $p(z)$-elliptic problems with a Hardy potential. Mathematical Modelling and Analysis, 29(3), 460-479(2024).
\end{thebibliography}
Published
2025-12-20
Section
Advances in Nonlinear Analysis and Applications
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