Existence results for nonlinear anisotropic elliptic partial differential equations with variable exponents
DOI:
https://doi.org/10.24193/subbmath.2025.3.11Keywords:
Nonlinear elliptic problem, growth conditions, anisotropic Sobolev spaces, variable exponents, distributional solution, existenceAbstract
The focus of this paper will be on studying the existence of solutions in the sense of distribution, for a class of nonlinear partial differential equations defined by a variable exponent anisotropic elliptic operator with a growth condition given by a strictly positive continuous real function. The functional setting involves variable exponents anisotropic Sobolev spaces.
Mathematics Subject Classification (2010): 35J60, 35J66, 35J67.
Received 28 January 2025; Accepted 10 June 2025.
References
, A., Harjulehto, P., Hästö, P., Lukkari, T., Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces, Ann. Mat. Pura. Appl., 194(2015), 405–424.
Bendahmane, M., Karlsen, KH., Anisotropic nonlinear elliptic systems with measure data and anisotropic harmonic maps into spheres, Electron. J. Differ. Equ., 46(2006), 1–30.
Chen, Y., Levine, S., Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66(2006), 1383–1406.
Cruz-Uribe, D., Fiorenza, A., Ruzhansky, M., Wirth, J., Variable Lebesgue Spaces and Hyperbolic Systems. Advanced Courses in Mathematics- CRM Barcelona. Birkhäuser, Basel, 2014.
Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Springer. vol. 2017, New York (2011).
Fan, X., Zhao, D., On the spaces Lp(x)(Ω) and W1,p(x)(Ω), J. Math. Anal. Appl., 263(2001), 424–446.
Fan, X., Anisotropic variable exponent Sobolev spaces and −→p (x)-Laplacian equations, Complex Var. Elliptic Equ., 56(2011), 623–642.
Lions, J. L., Quelques méthodes de résolution des problèmes aux limites, Dunod, Paris, 1969.
Mihailescu, M., Radulescu, V., A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. A., 462(2006), 2625–2641.
Naceri, M., Existence results for anisotropic nonlinear weighted elliptic equations with variable exponents and L1 data, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci., 23(4), pp. 337–346, 2022.
Naceri, M., Mokhtari, F., Anisotropic nonlinear elliptic systems with variable exponents and degenerate coercivity, Appl. Anal., 100(11)(2021), 2347–2367.
Naceri, M., Benboubker, M. B., Distributional solutions of anisotropic nonlinear elliptic systems with variable exponents: existence and regularity, Adv. Oper. Theory., 7(2)(2022), 1–34.
Naceri, M., Anisotropic nonlinear elliptic systems with variable exponents, degenerate coercivity and Lq(·) data, Ann. Acad. Rom. Sci. Ser. Math. Appl., 14(1-2)(2022), 107– 140.
Naceri, M., Anisotropic nonlinear weighted elliptic equations with variable exponents, Georgian Math. J., vol. 30, no. 2, 2023, pp. 277–285.
Naceri, M., Anisotropic nonlinear elliptic equations with variable exponents and two weighted first order terms, Filomat, 38(3)(2024), 1043–1054.
Naceri, M., Nonlinear Elliptic Equations with Variable Exponents Anisotropic Sobolev Weights and Natural Growth Terms, Tatra Mt. Math. Publ., 88(2024), 109–126.
Naceri, M., Entropy solutions for variable exponents nonlinear anisotropic elliptic equations with natural growth terms, Rev. Colombiana Mat., Vol. 58 No. 1 (2024), 99–115.
Naceri, M., Singular Anisotropic Elliptic Problems with Variable Exponents, Mem. Differ. Equ. Math. Phys., 85(8)(2022), 119 –132.
Naceri, M., Variable exponents anisotropic nonlinear elliptic systems with L→−p ′(·)−data. Appl. Anal., 104(3), 564–581(2024).
Naceri, M., Existence results for a nonlinear variable exponents anisotropic elliptic problems, Kyungpook Math. J., 64(2024), no. 2, 271–286.
Ružička, M., Electrorheological fluids: modelling and mathematical theory, Springer, Berlin. Lecture Notes in Mathematics,1748(2000).
Sanchón, M., Urbano, M., Entropy solutions for the p(x)-Laplace equation, Trans. Amer. Math. Soc., 361(2009), 6387–6405.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution 4.0 International License.
