On Choquard-Kirchhoff Type Critical Multiphase Problem

In this paper, we obtain the existence of weak solutions to the Choquard-Kirchhoff type critical multiphase problem: \begin{equation*} \left{\begin{array}{cc} &-M(\varphi_{\h}(\lvert{\nabla u}\rvert))div(\lvert{\nabla u}\rvert^{p(x)-2}\nabla u+a_1(x)\lvert{\nabla u}\rvert^{q(x)-2}\nabla u+a_2(x)\lvert{\nabla u}\rvert^{r(x)-2}\nabla u) & =λg(x)\lvert{u}\rvert^{γ(x)-2}u+θB(x,u)+κ\left(\int_{\q}\frac{F(y,u(y))}{\lvert{x-y}\rvert^{d(x,y)}}, dy\right) f(x,u) \ $\text{in}$ \ Ω, & u=0 \ $\text{on}$ \ {\partial Ω}. \end{array}\right. \end{equation*} The term $B(x,u)$ on the right-hand side generalizes the critical growth. We obtain existence and multiplicity results by establishing certain embedding results and concentration compactness principle along with the Hardy-Littlewood-Sobolev type inequality for the Musielak Orlicz Sobolev space $W^{1,\mathcal{T}}(\text{q})$.