Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent

Abstract

This paper is dedicated to studying the Choquard equation \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+V(x)u = (I_{\alpha}\ast|u|^{p})|u|^{p-2}u+g(u),\; \; \; \; \; x\in\mathbb{R}^{N}, u\in H^{1}(\mathbb{R}^{N}) \end{array} \right. \end{equation*} $\end{document} where \begin{document}$ N\geq4 $\end{document} , \begin{document}$ \alpha\in(0, N) $\end{document} , \begin{document}$ V\in\mathcal{C}(\mathbb{R}^{N}, \mathbb{R}) $\end{document} is sign-changing and periodic, \begin{document}$ I_{\alpha} $\end{document} is the Riesz potential, \begin{document}$ p = \frac{N+\alpha}{N-2} $\end{document} and \begin{document}$ g\in\mathcal{C}(\mathbb{R}, \mathbb{R}) $\end{document} . The equation is strongly indefinite, i.e., the operator \begin{document}$ -\Delta+V $\end{document} has infinite-dimensional negative and positive spaces. Moreover, the exponent \begin{document}$ p = \frac{N+\alpha}{N-2} $\end{document} is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality. Under some mild assumptions on \begin{document}$ g $\end{document} , we obtain the existence of nontrivial solutions for this equation.