Multiple positive bound state solutions of a critical Choquard equation

Abstract

IIn this paper we consider the problem $$ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u=(I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N},\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right.\leqno{(P_{\lambda})} $$ where $V_{\lambda}=\lambda+V_{0}$ with $\lambda \geq 0$, $V_0\in L^{N/2}(\R^N)$, $I_{\mu}=\frac{1}{|x|^\mu}$ is the Riesz potential with $0<\mu<\min\{N,4\}$ and $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$ with $N\geq 3$. Under some smallness assumption on $V_0$ and $\lambda$ we prove the existence of two positive solutions of $(P_\lambda)$. In order to prove the main result, we used variational methods combined with degree theory.