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Anisotropic nonlinear weighted elliptic equations with variable exponents

  • Mokhtar Naceri EMAIL logo
Published/Copyright: January 27, 2023
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Abstract

In this paper we prove the existence of distributional solutions to certain anisotropic nonlinear weighted elliptic equations with variable exponents, where the weight function belongs to the anisotropic Sobolev space with variable exponents and zero boundary. The functional setting involves anisotropic variable exponents Lebesgue–Sobolev spaces.

MSC 2010: 35J60; 35J66

Acknowledgements

The author would like to thank the referees for their comments and suggestions.

References

[1] A. Aberqi, J. Bennouna, O. Benslimane and M. A. Ragusa, Existence results for double phase problem in Sobolev–Orlicz spaces with variable exponents in complete manifold, Mediterr. J. Math. 19 (2022), no. 4, Paper No. 158. 10.1007/s00009-022-02097-0Search in Google Scholar

[2] R. P. Agarwal, O. Bazighifan and M. A. Ragusa, Nonlinear neutral delay differential equations of fourth-order: Oscillation of solutions, Entropy 23 (2021), no. 2, Paper No. 129. 10.3390/e23020129Search in Google Scholar PubMed PubMed Central

[3] R. P. Agarwal, S. Gala and M. A. Ragusa, A regularity criterion in weak spaces to Boussinesq equations, Mathematics 8 (2020), no. 6, Paper No. 920. 10.3390/math8060920Search in Google Scholar

[4] A. M. Alghamdi, S. Gala, C. Qian and M. A. Ragusa, The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations, Electron. Res. Arch. 28 (2020), no. 1, 183–193. 10.3934/era.2020012Search in Google Scholar

[5] H. Ayadi and F. Mokhtari, Nonlinear anisotropic elliptic equations with variable exponents and degenerate coercivity, Electron. J. Differential Equations 2018 (2018), Paper No. 45. Search in Google Scholar

[6] M. Bendahmane, K. H. Karlsen and M. Saad, Nonlinear anisotropic elliptic and parabolic equations with variable exponents and L 1 data, Commun. Pure Appl. Anal. 12 (2013), no. 3, 1201–1220. 10.3934/cpaa.2013.12.1201Search in Google Scholar

[7] M. Bendahmane and F. Mokhtari, Nonlinear elliptic systems with variable exponents and measure data, Moroc. J. Pure Appl. Anal. 1 (2015), no. 2, 108–125. 10.7603/s40956-015-0008-3Search in Google Scholar

[8] O. Benslimane, A. Aberqi and J. Bennouna, Existence and uniqueness of weak solution of p ( x ) -Laplacian in Sobolev spaces with variable exponents in complete manifolds, Filomat 35 (2021), no. 5, 1453–1463. 10.2298/FIL2105453BSearch in Google Scholar

[9] L. Boccardo, P. Imparato and L. Orsina, Nonlinear weighted elliptic equations with Sobolev weights, Boll. Unione Mat. Ital. 15 (2022), no. 4, 503–514. 10.1007/s40574-021-00314-4Search in Google Scholar

[10] L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (1992), no. 6, 581–597. 10.1016/0362-546X(92)90023-8Search in Google Scholar

[11] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406. 10.1137/050624522Search in Google Scholar

[12] D. Cruz-Uribe, A. Fiorenza, M. Ruzhansky and J. Wirth, Variable Lebesgue Spaces and Hyperbolic Systems, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Basel, 2014. 10.1007/978-3-0348-0840-8Search in Google Scholar

[13] J. Dai, A Steiner inequality for the anisotropic perimeter, J. Funct. Spaces 2022 (2022), Article ID 2989121. 10.1155/2022/2989121Search in Google Scholar

[14] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011. 10.1007/978-3-642-18363-8Search in Google Scholar

[15] X. Fan, Anisotropic variable exponent Sobolev spaces and p ( x ) -Laplacian equations, Complex Var. Elliptic Equ. 56 (2011), no. 7–9, 623–642. 10.1080/17476931003728412Search in Google Scholar

[16] X. Fan and D. Zhao, On the spaces L p ( x ) ( Ω ) and W m , p ( x ) ( Ω ) , J. Math. Anal. Appl. 263 (2001), no. 2, 424–446. 10.1006/jmaa.2000.7617Search in Google Scholar

[17] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover, Mineola, 2006. Search in Google Scholar

[18] M. Mihăilescu and V. Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2073, 2625–2641. 10.1098/rspa.2005.1633Search in Google Scholar

[19] N. Mokhtar, Anisotropic nonlinear elliptic systems with variable exponents, degenerate coercivity and L q ( ) data, Ann. Acad. Rom. Sci. Ser. Math. Appl. 14 (2022), no. 1–2, 107–140. 10.56082/annalsarscimath.2022.1-2.107Search in Google Scholar

[20] N. Mokhtar and M. B. Benboubker, Distributional solutions of anisotropic nonlinear elliptic systems with variable exponents: existence and regularity, Adv. Oper. Theory 7 (2022), no. 2, Paper No. 17. 10.1007/s43036-022-00183-4Search in Google Scholar

[21] N. Mokhtar and F. Mokhtari, Anisotropic nonlinear elliptic systems with variable exponents and degenerate coercivity, Appl. Anal. 100 (2021), no. 11, 2347–2367. 10.1080/00036811.2019.1682136Search in Google Scholar

[22] F. Mokhtari, Nonlinear anisotropic elliptic equations in N with variable exponents and locally integrable data, Math. Methods Appl. Sci. 40 (2017), no. 6, 2265–2276. 10.1002/mma.4137Search in Google Scholar

[23] F. Mokhtari, Regularity of the solution to nonlinear anisotropic elliptic equations with variable exponents and irregular data, Mediterr. J. Math. 14 (2017), no. 3, Paper No. 141. 10.1007/s00009-017-0941-7Search in Google Scholar

[24] M. Naceri, Singular anisotropic elliptic problems with variable exponents, Mem. Differ. Equ. Math. Phys. 85 (2022), 119–132. Search in Google Scholar

[25] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. 1748, Springer, Berlin, 2000. 10.1007/BFb0104029Search in Google Scholar

Received: 2022-06-18
Revised: 2022-07-14
Accepted: 2022-07-22
Published Online: 2023-01-27
Published in Print: 2023-04-01

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