International Journal of Nonlinear Analysis and Applications

On the existence of a solution for a strongly nonlinear elliptic perturbed anisotropic problem of infinite order with variable exponents

Document Type : Research Paper

Author

Equipe EDP et Calcul Scientifique, Laboratoire de Mathematiques et Leurs Interactions, Faculte des Sciences, Moulay Ismail University, Meknes, Morocco

Abstract

In this work, we shall be interested in the existence of a solution to the following Dirichlet problem for a specific class of elliptical anisotropic equations of the type
\begin{eqnarray}\label{P.1}
\left \{\begin{array}{rl}
&A(u)+g(x,u)= f \ \ {\rm in}\
\Omega
%[1.5ex]
\\
%[1ex]
& u=0\ \ {\rm on}\ {\partial \Omega},
\end{array}
\right.
\end{eqnarray}
where $\Omega$ is a bounded open set of $\mathbb{R}^{N},$ $A=\sum_{|\alpha|=0}^{\infty}(-1)^{|\alpha|}D^{\alpha}\big(a_{\alpha}|D^{\alpha}u|^{p_{\alpha}(x)-2}D^{\alpha}u\big)$ is an operator of infinite order and $g(x, s )$ is a non-linear lower order term that verify some natural growth and sign conditions, where the data $f$ is framed in $L^1(\Omega)$.

Keywords

[1] M.H. Abdou, A. Benkirane M. Chrif, and S. El Manouni, Strongly anisotropic elliptic problems of infinite order with variable exponents, Complex Var. Elliptic Equ. 59 (2014), no. 10, 1403–1417.
[2] E. Acerbi and G. Mingione, Gradient estimates for the p(x)-Laplacian system, J. Reine Angew. Math. 584 (2005), 117–148.
[3] R. Adams, Sobolev Spaces, Press New York, 1975.
[4] C.O. Alves and M.A. Souto, Existence of solutions for a class of problems in IRN involving the p(x)-Laplacian, T. Cazenave, D. Costa, O. Lopes, R. Manasevich, P. Rabinowitz, B. Ruf, C. Tomei (Eds.), Contributions to Nonlinear Analysis, A Tribute to D.G. de Figueiredo on the Occasion of his 70th Birthday, Progr. Nonlinear Differential Equations Appl., vol. 66, Birkh¨auser, Basel, 2006, pp. 17–32.
[5] A. Benkirane M. Chrif, and S. El Manouni, Existence results for strongly nonlinear equations of infinite order, Z. Anal. Anwend. (J. Anal. Appl.) 26 (2007), 303–312.
[6] H. Brezis, Analyse fonctionelle: Theorie, Methodes et Applications, Masson, Paris, 1992.
[7] J. Chabrowski and Y. Fu, Existence of solutions for p(x)-Laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), 604–618.
[8] M. Chrif and S. El Manouni, On a strongly anisotropic equation with L1-data, Appl. Anal. 87 (2088), no. 7, 865–871.
[9] M. Chrif and S. El Manouni, Anisotropic equations in weighted Sobolev spaces of higher order, Ricerche Mat. 58 (2009), 1–14.
[10] L. Diening, Theoretical and numerical results for electrorheological fluids, PhD thesis, University of Freiburg, Germany, 2002.
[11] L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin, 2011.
[12] J.A. Dubinskii, Sobolev spaces for infinite order and the behavior of solutions of some boundary value problems with unbounded increase of the order of the equation, Math. USSR-Sb. 27 (1975), no. 2, 143–162.
[13] J.A. Dubinskii, Sobolev Spaces of Infinite Order and Differential Equations, Teubner-Texte Math., Band 87. Leipzig: Teubner, 1986.
[14] D.E. Edmunds, J. Lang, and A. Nekvinda, On Lp(x) norms, Proc. Roy. Soc. London Ser. A 455 (1999), 219–225.
[15] D.E. Edmunds and J. Rakosnik, Density of smooth functions in Wk,p(x)(Ω), Proc. Roy. Soc. London Ser. A 437 (1992), 229–236.
[16] D.E. Edmunds and J. Rakosnık, Sobolev embedding with variable exponent, Studia Math. 143 (2000), 267–293.
[17] X.L. Fan, D. Zhao, On the generalized Orlicz-Sobolev space Wk,p(x)(Ω), J. Gansu Educ. College 12 (1998), no. 1, 1–6.
[18] P. Harjulehto, P. Hasto, U. V. Le, and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), 4551–4574.
[19] T.C. Halsey, Electrorheological fluids, Science 258 (1992), 761–766.
[20] S. Heidari and A. Razani, Multiple solutions for a class of nonlocal quasilinear elliptic systems in Orlicz-Sobolev spaces, Bound. Value Prob. 2021 (2021), no.22, 15 pages.
[21] A. Khaleghi and A. Razani, Solutions to a (p(x); q(x))-biharmonic elliptic problem on a bounded domain, Bound. Value Prob. 2023 (2023), 53.
[22] O. Kovacik and J. Rakosnik, On spaces Lp(x) and W1,p(x), Czech. Math. J. 41 (1991), 592–618.
[23] J.L. Lions, Quelque Methodes de Resolution des Problemes aux Limites Non Lineaires, Paris: Dunod, Gauthier- Villars 1969.
[24] M. Mihailescu, V. Radulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. Roy. Soc. London Ser. A 462 (2006), 2625–2641.
[25] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer-Verlag, Berlin, 1983.
[26] C. Pfeiffer, C.Mavroidis, Y. Bar-Cohen, and B. Dolgin, Electrorheological fluid based force feedback device, Proc. 1999 SPIE Telemanipulator and Telepresence Technologies VI Conf., vol. 3840, Boston, MA, 1999.
[27] A. Razani, Entire weak solutions for an anisotropic equation in the Heisenberg group, Proc. Amer. Math. Soc. 151 (2023), no. 11.
[28] A. Razani, Non-existence of solution of Haraux-Weissler equation on a strictly star-shaped domain, Miskolc Math. Notes 24 (2023), no. 1, 395–402,
[29] A. Razani and G.M. Figueiredo, A positive solution for an anisotropic p£q-Laplacian, Discrete Contin. Dyn. Syst. Ser. S 16 (2023), no. 6, 1629–1643.
[30] S. Samko and B. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005), 229-246.
[31] D. Zhao, W.J. Qiang, and X.L. Fan, On generalized Orlicz spaces Lp(x), J. Gansu Sci. 9 (1996), no. 2, 1–7.
International Journal of Nonlinear Analysis and Applications
Volume 16, Issue 3
March 2025
Pages 53-62
  • Receive Date: 20 December 2023
  • Revise Date: 17 January 2024
  • Accept Date: 19 January 2024