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

Abstract

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_Ω\mathcal{H}(x,|\nabla u|)dx \right) Δ_{\mathcal{H}}u &=f(x,u) \ \ \hbox{in} \ \ Ω, \ \ \ \\ \hspace{3.3cm} u &= 0 \ \ \hbox{on} \ \ \partial Ω, \end{array}\right. \end{equation*} where $Ω\subseteq \mathbb{R}^{N},\, N\geq 2$ is a bounded and smooth domain containing two open and connected subsets $Ω_p$ and $Ω_N$ such that $ \barΩ_{p}\cap\barΩ_{N}=\emptyset$ and $Δ_{\mathcal{H}}u=\hbox{div}( h(x,|\nabla u|)\nabla u)$ is the $\mathcal{H}$-Laplace operator. We assume that $Δ_{\mathcal{H}}$ reduces to $ Δ_{p(x)}$ in $Ω_{p}$ and to $ Δ_{N}$ in $Ω_{N},$ the non-linear function $f:Ω\times\mathbb{R}\rightarrow \mathbb{R}$ act as $|t|^{p^{\ast}(x)-2}t$ on $Ω_{p}$ and as $e^{α|t|^{N/(N-1)}}$ on $Ω_{N}$ for sufficiently large $|t|$. To establish our existence results in a Musielak-Sobolev space, we use a variational technique based on the mountain pass theorem.