Ground state solutions for elliptic Kirchhoff–Boussinesq type problems with supercritical exponential growth

In this work, we are interested in studying the existence of a ground state solution for the problem $$ \Delta (w_{\beta}(x)\Delta u ) \pm \text{div}(w_{\beta} (x) | \nabla u |^{p-2} \nabla u)= f(x,u) \text{ in }B,\quad \text{and} \ u= \dfrac{\partial u}{\partial \nu}=0 \ \text{on}\ \partial B, $$ where B is the unit ball in $\mathbb{R}^{4}$ , $w_{\beta}(x)=\big(\log \frac{e}{|x|}\big)^{\beta}$ or $w_{\beta}(x)=\big(\log (\frac{1}{|x|}\big)^{\beta}$ for $\beta \in (0,1)$ , $2< p < 4$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous class function with supercritical exponential growth. We utilize the Nehari manifold method to establish the existence result.