Kirchhoff type elliptic equations with double criticality in Musielak-Sobolev spaces

This paper aims to establish the existence of a weak solution for the non-local problem: \begin{equation*} \left{\begin{array}{ll} -a\left(\int_{\Omega}$\mathcal{H}$(x,|\nabla u|)dx \right) \Delta_{$\mathcal{H}$}u &=f(x,u) \ \ \hbox{in} \ \ \Omega, \ \ \ \ \hspace{3.3cm} u &= 0 \ \ \hbox{on} \ \ \partial \Omega, \end{array}\right. \end{equation*} where $\Omega\subseteq $\mathbb{R}$^{N},, N\geq 2$ is a bounded and smooth domain containing two open and connected subsets ${\mathrm{\Omega }}_p$ and ${\mathrm{\Omega }}_N$ such that ${\bar{\mathrm{\Omega }}}_p\cap {\bar{\mathrm{\Omega }}}_N=\mathrm{\varnothing }$ and ${\mathrm{\Delta }}_{\mathcal{H}}u=\text{div}(h(x,\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })\mathrm{\nabla }u)$ is the $\mathcal{H}$-Laplace operator. We assume that ${\mathrm{\Delta }}_{\mathcal{H}}$ reduces to ${\mathrm{\Delta }}_{p(x)}$ in ${\mathrm{\Omega }}_p$ and to ${\mathrm{\Delta }}_N$ in ${\mathrm{\Omega }}_N,$ the non-linear function $f:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ act as $\mathrm{\mid }t{\mathrm{\mid }}^{p^{\ast }(x)-2}t$ on ${\mathrm{\Omega }}_p$ and as $e^{\alpha \mathrm{\mid }t{\mathrm{\mid }}^{N\mathrm{/}(N-1)}}$ on ${\mathrm{\Omega }}_N$ for sufficiently large $\mathrm{\mid }t\mathrm{\mid }$. To establish our existence results in a Musielak-Sobolev space, we use a variational technique based on the mountain pass theorem.