Existence of solutions for a strongly nonlinear (p(x),q(x))-elliptic systems via topological degree

Document Type : Research Paper


1 Laboratory of Applied Mathematics and History and Didactics of Mathematics "LAMAHIS", Department of Mathematics, University of 20 August 1955, Skikda, Algeria

2 Faculty of Sciences, University of 20 August 1955, Skikda, Algeria



This article is concerned with the study of the existence of a distributional solution for a strongly nonlinear (p(x),q(x))-elliptic systems. By means of the Berkovits degree theory, with suitable assumptions on the nonlinearities, we prove the existence of nontrivial solutions to our problem.


[1] E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), no. 3, 213–259.
[2] M. Ait Hammou and E. Azroul, Existence result for a nonlinear elliptic problem by topological degree in Sobolev spaces with variable exponent, Moroccan J. Pure Appl. Anal. 7 (2021), no. 1, 50–65.
[3] M. Ait Hammou, E. Azroul, and B. Lahmi, Existence of solutions for p(x)-Laplacian Dirichlet problem by Topological degree, Bull. Transilv. Univ. Brasov Ser III. 11 (2018), no. 2, 29–38.
[4] M. Ait Hammou, E. Azroul, and B. Lahmi, Topological degree methods for a Strongly nonlinear p(x)−elliptic problem, Rev. Colombiana Mat. 53 (2019), no. 1, 27–39.
[5] J. Berkovits, Extension of the Leray-Schauder degree for abstract Hammerstein type mappings, J. Differ. Equ. 234 (2007), no. 1, 289–310.
[6] L.E.J. Brouwer, Uber Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1912), 97–115.
[7] F.E. Browder, Degree of mapping for nonlinear mappings of monotone type, Proc. Natl. Acad. Sci. USA. 80 (1983), no. 6, 1771–1773.
[8] L. Dingien, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev Spaces with Vriable Exponent, Lecture Notes in Mathematics, Springer, Berlin, 2011.
[9] G.G. dos Santos, G.M. Figueiredo, and L.S. Tavares, Sub-super solution method for nonlocal systems involving the p(x)−Laplacian operator, Electron. J. Differ. Equ. 2020 (2020), no. 25, 1–19.
[10] D.E. Edmunds, J. Lang, and A. Nekvinda, On Lp(x)(Ω) norms, Proc. Royal Soc. London Ser. A 455 (1999), 219–225.
[11] D.E. Edmunds and J. Rakosnik, Sobolev embeddings with variable exponent, Studia Math. 143 (2000), no. 3, 267–293.
[12] X. Fan, Q. Zhang, and D. Zhao, Eigenvalues of p(x)−Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), no. 2, 306–317.
[13] X.L. Fan, J. Shen, and D. Zhao, Sobolev embedding theorems for spaces Wm,p(x)(Ω), J. Math. Anal. Appl. 262 (2001), no. 2, 749–760.
[14] X.L Fan and Q.H. Zhang, Existence of solutions for p(x)−Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), no. 8, 1843–1852.
[15] X.L. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), no. 2, 424–446.
[16] G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl. 367 (2010), 204–228.
[17] J. Giacomoni and G. Vallet, Some results about an anisotropic p(x)-Laplace Barenblat equation, Adv. Nonlinear Anal. 1 (2012), 227–298.
[18] S. Heidari and A. Razani, Infinitely many solutions for (p(x); q(x))- Laplacian-like systems, Commun. Korean Math. Soc. 36 (2021), no. 1, 51–62.
[19] F. Karami, K. Sadik, and L. Ziad A variable exponent nonlocal p(x)−Laplacian equation for image restoration, Comput. Math. Appl. 75 (2018), 534–546.
[20] 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.
[21] I.S. Kim and S.J. Hong, A topological degree for operators of generalized (S+) type, Fixed Point Theory Algorithms Sci. Eng. 2015 (2015), 194.
[22] O. Kovacik and J. Rakosnik, On spaces Lp(x)(Ω) and Wm,p(x)(Ω), Czechoslovak Math. J. 41 (1991), 592–618.
[23] H. Lalilia, S. Tasa, and A. Djellitb, Existence of solutions for critical systems with variable exponents, Math. Modell. Anal. 23 (2018), 596–610.
[24] S. Lecheheb and A. Fekrache, Topological degree methods for a nonlinear elliptic systems with variable exponents, Stud. Univ. Babes-Bolyai Math., In Press.
[25] J. Leray and J. Schauder, Topologie et equations fonctionnelles, Ann. Sci. Ec. Norm. Super 51 (1934), no. 3, 45–78.
[26] N. Mokhtar and F. Mokhtari, Anisotropic nonlinear elliptic systems with variable exponents and degenerate coercivity, Appl. Anal. 100 (2019), no. 11, 2347–2367.
[27] A. Moussaoui and J. Velin, Existence and a priori estimates of solutions for quasilinear singular elliptic systems with variable exponents, J. Elliptic Parabol. Equ. 4 (2018), 417–440.
[28] V. Radulescu and D. Repovs, Partial Differential Equations with Variable Exponents, Variational Methods and Qualitative Analysis, Monographs and Research Note in Mathematics. CRC Press, Boca Raton, FL, 2015.
[29] A. Razani, Two weak solutions for fully nonlinear Kirchhoff-type problem, Filomat 35 (2021), no. 10, 3267–3278.
[30] A. Razani and F. Safari, A (p(.); q(.))-Laplacian problem with the Steklov boundary conditions, Lobachevskii J. Math. 43 (2022), no. 12, 3616–3625.
[31] 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.
[32] O. Saifia and J. VelinK, J., Existence result for variable exponents elliptic system with lack of compactness, Appl. Anal., 101 (2020), no. 6, 2119-2143.
[33] F. Souilah, M. Maouni, and K. Slimani, Quasilinear parabolic problems in the Lebsgue-Sobolev space with variable exponent and L1 data, Int. J. Nonlinear Anal. Appl., In Press, 1-15. 10.22075/ijnaa.2023.30528.4423
[34] N. Tsouli and O. Darhouche, Existence and multiplicity results for nonlinear problems involving the p(x)-Laplace operator, Opuscula Math. 34 (2014), no. 3, 621–638.
[35] L. Yin, Y. Liang, Q. Zhang, and C. Zhao, Existence of solutions for a variable exponent system without PS conditions, Electronic J. Differ. Equ. 2015 (2015), no. 63, 1–23.
[36] E. Zeidler, Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators, Springer, New York, 1990.
[37] Q. Zhang and C. Zhao, Existence of strong solutions of a p(x)−Laplacian Dirichlet problem without the Ambrosetti–Rabinowitz condition, Comput. Math. Appl. 69 (2015), 1–12.
[38] Q. Zhang, Y. Guo, and G. Chen, Existence and multiple solutions for a variable exponent system, Nonlinear Anal.: Theory Meth. Appl. 73 (2010), 3788–3804.
[39] D. Zhao, W.J. Qiang, and X.L. Fan, On generalized Orlicz spaces Lp(x)(Ω), J. Gansu Sci. 9 (1996), no. 2, 1–7.
[40] V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. 29 (1987), no. 1, 33–66.

Articles in Press, Corrected Proof
Available Online from 03 April 2024
  • Receive Date: 07 January 2024
  • Revise Date: 16 February 2024
  • Accept Date: 19 February 2024