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

This paper aims to establish the existence of a weak solution for 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 \end{array}\right. \end{equation*} where $\Omega\subseteq \mathbb{R}^{N},\, N\geq 2$ is bounded and smooth domain containing two open connected subsets $\Omega_p$ $\Omega_N$ such that $ \bar{\Omega}_{p}\cap\bar{\Omega}_{N}=\emptyset$ $\Delta_{\mathcal{H}}u=\hbox{div}( h(x,|\nabla u|)\nabla u)$ $\mathcal{H}$-Laplace operator. We assume $\Delta_{\mathcal{H}}$ reduces \Delta_{p(x)}$ in $\Omega_{p}$ \Delta_{N}$ $\Omega_{N},$ non-linear function $f:\Omega\times\mathbb{R}\rightarrow \mathbb{R}$ act as $|t|^{p^{\ast}(x)-2}t$ on $e^{\alpha|t|^{N/(N-1)}}$ $\Omega_{N}$ sufficiently large $|t|$. To our results Musielak-Sobolev space, we use variational technique based mountain pass theorem.