Investigating the existence and multiplicity of solutions to \(\varphi(x)\)-Kirchhoff problem

In this article, we want to discuss variational methods such as the Mountain pass theorem and the Symmetric Mountain pass theorem, without the Ambrosetti-Rabinowitz condition. We prove the existence and multiplicity of nontrivial weak solutions for the problem of the following form \[ \begin{align*} \begin{cases} -\left(\alpha-\beta \displaystyle\int_\Omega \frac{1}{\varphi(x)} |\nabla \upsilon|^{\varphi(x)} \, dx\right) \Delta_{\varphi(x)} \upsilon + |\upsilon|^{\psi(x)-2}\upsilon \\ \hfill = \lambda \eta(x, \upsilon), & x \in \Omega, \\ \left(\alpha-\beta \displaystyle\int_{\partial \Omega} \frac{1}{\varphi(x)} |\nabla \upsilon|^{\varphi(x)} \, dx\right) |\nabla \upsilon|^{\varphi(x)-2} \frac{\partial \upsilon}{\partial \nu} = 0, & x \in \partial \Omega, \end{cases} \end{align*} \] where \(\alpha \geq \beta > 0\), \(\Delta_{\varphi(x)} \upsilon\) is the \(\varphi(x)\)-Laplacian operator, \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial \Omega\), and \(\nu\) is the outer unit normal to \(\partial \Omega\). Additionally, \(\varphi(x), \psi(x) \in C(\bar{\Omega})\) with \(1 < \varphi(x) < N,~ \varphi(x) < \psi(x) < \varphi^*(x) := \frac{N \varphi(x)}{N- \varphi(x)}\), \(\lambda > 0\) is a real parameter, and \(\eta(x, t) \in C(\bar{\Omega} \times \mathbb{R}, \mathbb{R})\).