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

Document Type : Research Paper

Authors

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

10.22075/ijnaa.2023.32433.4823

Abstract

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
\begin{equation*}
\left\{\begin{array}{ll}
-\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.
\end{equation*}
The methodology adopted revolves around the technical approach utilizing the direct variational method within the framework of anisotropic variable exponent Sobolev spaces.

Keywords

[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. https://doi.org/10.2989/16073606.2023.2190045
[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