The Entropy Solutions for Anisotropic Neumann Problems with Variable Exponents and Hardy Potential
Entropy Solutions for Anisotropic Neumann Problems
DOI:
https://doi.org/10.5269/bspm.78871Resumo
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.
Referências
\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}
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2025-12-20
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Conf. Issue: Advances in Nonlinear Analysis and Applications
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