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

Abstract

<p style='text-indent:20px;'>We are concerned with the existence of ground state solutions to the nonhomogeneous perturbed Choquard equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ - \Delta_{p(x)} u + V(x)|u|^{p(x) - 2} u $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ = \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, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where the exponent <inline-formula><tex-math id="M1">\begin{document}$ r(\cdot) $\end{document}</tex-math></inline-formula> is critical with respect to the Hardy-Littlewood-Sobolev inequality for variable exponents. We first consider the case where the perturbation <inline-formula><tex-math id="M2">\begin{document}$ g(\cdot ,\cdot) $\end{document}</tex-math></inline-formula> is subcritical and we distinguish between the superlinear and sublinear cases. In both situations we establish the existence of solutions and we prove the asymptotic behavior of low-energy solutions in the case of high perturbations. Next, we study the case where the nonlinearity <inline-formula><tex-math id="M3">\begin{document}$ g(\cdot ,\cdot) $\end{document}</tex-math></inline-formula> is critical. We prove the existence of solutions both for low and high perturbations and we establish asymptotic properties of low-energy solutions.</p>