Multiplicity analysis of positive weak solutions in a quasi-linear Dirichlet problem inspired by Kirchhoff-type phenomena

Document Type : Research Paper


1 {Mathematics and Computer Sciences Department, Research Unit Geometry, Algebra, Analysis and Applications, Faculty of Science and Technology, University of Nouakchott, Nouakchott, Mauritania

2 Department of Industrial Engineering and Applied Mathematics, Professional University Institute, University of Nouakchott, Nouakchott, Mauritania



The main focus of this paper lies in investigating the existence of infinitely many positive weak solutions for the following elliptic-Kirchhoff equation with Dirichlet boundary condition
-\sum_{i=1}^{N}M_{i}\left(\int_{\Omega}\displaystyle\frac{1}{p_{i}(x)}\displaystyle\Big|\frac{\partial u}{\partial x_{i}}\Big|^{p_{i}(x)}dx\right)\frac{\partial}{\partial x_{i}}\left(\Big|\frac{\partial u}{\partial x_{i}}\Big|^{p_{i}(x)-2}
\frac{\partial u}{\partial x_{i}}\right) = f(x,u) &\mbox{ in } \Omega, \\
u =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right.
The methodology adopted revolves around the technical approach utilizing the direct variational method within the framework of anisotropic variable exponent Sobolev spaces.


[1] A. Ahmed, H. Hjiaj, and A. Touzani, Existence of infinitely many weak solutions for a Neumann elliptic equations involving the ⃗p(・)-Laplacian operator, Rend. Circ. Mat. Palermo (2) 64 (2015), no. 3, 459–473.
[2] M. Allaoui and A. Ourraoui, Existence results for a class of p(x)-Kirchhoff problem with a singular weight, Mediterr. J. Math. 13 (2016), no. 2, 677–686.
[3] C.O. Alves and A. El Hamidi, Existence of solution for an anisotropic equation with critical exponent, Nonlinear Anal. TMA. 4 (2005), 611–624.
[4] A.E. Amrouss and A. El Mahraoui, Infinitely many solutions for anisotropic elliptic equations with variable exponent, Proyec. J. Math. 40 (2021), no. 5, 1071–1096.
[5] S.N. Antontsev and J.F. Rodrigues, On stationary thermorheological viscous flows, Ann. Univer. Ferrara. 52 (2006), no. 1, 19–36.
[6] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), no. 1, 305–330.
[7] M. Avci, R.A. Mashiyev, and B. Cekic, Solutions of an anisotropic nonlocal problem involving variable exponent, Adv. Nonlinear Anal. 2 (2013), no. 3, 325–338.
[8] M.M. Boureanu, A. Matei, and M. Sofonea, Non-linear problems with p(・)-growth conditions and applications to anti-plane contact models, Adv. Nonlinear Stud. 14 (2014), no. 2, 295–313.
[9] F. Cammaroto and L. Vilasi, Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator, Nonlinear Anal. 74 (2011), no. 5, 1841–1852.
[10] Y. Chen, S. Levine, and R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406.
[11] N.T. Chung, Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities, Electron. J. Qual. Theory Differ. Equ. 42 (2012), 1–13.
[12] F.J.S.A. Correa and R.G. Nascimento, On a nonlocal elliptic system of p-Kirchhoff-type under Neumann boundary condition, Math. Model. Comput. 49 (2009), no. 3-4, 598–604.
[13] G. Dai and D. Liu, Infinitely many positive solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009), no. 2, 704–710. 
[14] G. Dai and R. Hao, Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009), no. 1, 275–284.
[15] L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, 2011.
[16] G.C.G. dos Santos, J.R.S. Silva, S.C.Q. Arruda, and L.S. Tavares, Existence and multiplicity results for critical anisotropic Kirchhoff-type problems with nonlocal nonlinearities, Complex Var. Elliptic Equ. 67 (2022), no. 4, 822–842.
[17] X.L. Fan, Anisotropic variable exponent Sobolev spaces and ⃗p(・)-Laplacian equations, Complex Var. Elliptic Equ. 56 (2011), no. 7-9, 623–642.
[18] X.L. Fan, On nonlocal p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 72 (2010), no. 7-8, 3314–3323.
[19] J.R. Graef, S. Heidarkhani, and L. Kong, A variational approach to a Kirchhoff-type problem involving two parameters, Results Math. 63 (2013), no. 3-4, 877–889.
[20] G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883.
[21] B. Kone, S. Ouaro, and S. Traore, Weak solutions for anisotropic nonlinear elliptic equations with variable exponents, Electron. J. Differ. Equ. 144 (2009), 1–11.
[22] D. Liu, On a p-Kirchhof equation via fountain theorem and dual fountain theorem, Nonlinear Anal. 72 (2010), no. 1, 302-308.
[23] M. Mihailescu and G. Morosanu, Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Appl. Anal. 89 (2010), no. 2, 257–271.
[24] M. Mihailescu, P. Pucci, and V. Radulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), no. 1, 687–698.
[25] A. Ourraoui, Multiplicity of solutions for p(.)-Laplacian elliptic Kirchhoff type equations, Appl. Math. E-Notes. 20 (2020), 124–132.
[26] J. Rakosnık, Some remarks to anisotropic Sobolev spaces I, Beitrage Anal. 13 (1979), 55-68.
[27] J. Rakosnık, Some remarks to anisotropic Sobolev spaces II, Beitrage Anal. 15 (1981), 127–140.
[28] A. Razani and G. M. Figueiredo, Degenerated and competing anisotropic (p, q)-Laplacians with weights, Appl. Anal. 102 (2023), no. 16, 4471–4488.
[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] A. Razani and G.M. Figueiredo, Existence of infinitely many solutions for an anisotropic equation using genus theory, Math. Meth. Appl. Sci. 45 (2022), no. 12, 7591-7606
[31] A. Razani, Nonstandard competing anisotropic (p, q)-Laplacians with convolution, Bound. Value Probl. 2022 (2022), 87.
[32] B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Glob. Optim. 46 (2010), no. 4, 543–549.
[33] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Springer, Berlin. 1748, 2000.
[34] T. Soltani and A. Razani, Solutions for an anisotropic elliptic problem involving nonlinear terms, Q. Math. (2023), 1-20.
[35] R. Stanway, J.L. Sproston, and A.K. El-Wahed, Applications of electro-rheological fluids in vibration control: A survey, Smart Mater. Struct. 5 (1996), no. 4, 464-482.
[36] V.V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR-Izvestiya 29 (1987), no. 1, 33–66.

Articles in Press, Corrected Proof
Available Online from 30 January 2024
  • Receive Date: 24 November 2023
  • Accept Date: 15 December 2023