High and low perturbations of Choquard equations with critical reaction and variable growth

We are concerned with the existence of ground state solutions to the nonhomogeneous perturbed Choquard equation

\begin{document}$ - \Delta_{p(x)} u + V(x)|u|^{p(x) - 2} u $\end{document}

\begin{document}$ = \left( \int_{\mathbb R^N} r(y)^{-1}|u(y)|^{r(y)}|x-y|^{-\lambda(x,y)} dy\right) |u|^{r(x)-2} u+g(x,u)\ \mbox{in}\ \mathbb R^N, $\end{document}

where the exponent \begin{document}$ r(\cdot) $\end{document} is critical with respect to the Hardy-Littlewood-Sobolev inequality for variable exponents. We first consider the case where the perturbation \begin{document}$ g(\cdot ,\cdot) $\end{document} is subcritical and we distinguish between the superlinear and sublinear cases. In both situations we establish the existence of solutions and we prove the asymptotic behavior of low-energy solutions in the case of high perturbations. Next, we study the case where the nonlinearity \begin{document}$ g(\cdot ,\cdot) $\end{document} is critical. We prove the existence of solutions both for low and high perturbations and we establish asymptotic properties of low-energy solutions.