International Journal of Nonlinear Analysis and Applications

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

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

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International Journal of Nonlinear Analysis and Applications
Volume 16, Issue 1
January 2025
Pages 359-369
  • Receive Date: 24 November 2023
  • Accept Date: 15 December 2023