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

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.

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

Nonlinear elliptic equations ، Variational methods applied on PDEs ، Positive solutions to PDEs