On Choquard-Kirchhoff type critical multiphase problem

This article focuses on the study of the following Choquard-Kirchhoff type critical multiphase problem: \begin{alignat*}2 -M& (\varphi_{\h} (\p{u}))\\ &\times\text{div} \big(\p{u}^{p(x)-2}\nabla u +a_1(x)\p{u}^{q(x)-2}\nabla u +a_2(x)\p{u}^{r(x)-2}\nabla u\big)\hidewidth \\ =&\e g(x)\ve{u}^{\gamma(x)-2}u+\theta B(x,u) \\ &+\kappa \left(\int_{\q}\frac{F(y,u(y))}{\ve{x-y}^{d(x,y)}} dy\right) f(x,u) &\quad\hskip2.6cm & \text{in } \q,\\ u&=0 &\quad & \text{on } {\partial \Omega}. \end{alignat*} Here, the nonlinearity $B(x,u)$ exhibits critical growth. To handle this critical growth, we present the concentration compactness principle in the space $ W_0^{1,\h}(\q)$. To address the Choquard term, we prove the Hardy-Littlewood-Sobolev-type inequality in the framework of the generalized Sobolev space $ W_0^{1,\h}(\q)$. These tools, combined with variational methods, are used to establish the existence and multiplicity of weak solutions.