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

Abstract

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.