Multiple positive bound state solutions for a critical Choquard equation

Abstract

<p style='text-indent:20px;'>In this paper we consider the problem <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (P_{\lambda})\ \ \ \ \ \ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u = (I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N}, \\ u&gt;0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right. $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ V_{\lambda} = \lambda+V_{0} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ \lambda \geq 0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ V_0\in L^{N/2}({\mathbb{R}}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ I_{\mu} = \frac{1}{|x|^\mu} $\end{document}</tex-math></inline-formula> is the Riesz potential with <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\mu&lt;\min\{N, 4\} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ 2^{*}_{\mu} = \frac{2N-\mu}{N-2} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ N\geq 3 $\end{document}</tex-math></inline-formula>. Under some smallness assumption on <inline-formula><tex-math id="M8">\begin{document}$ V_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> we prove the existence of two positive solutions of <inline-formula><tex-math id="M10">\begin{document}$ (P_\lambda) $\end{document}</tex-math></inline-formula>. In order to prove the main results, we used variational methods combined with degree theory.