Multiple positive bound state solutions of a critical Choquard equation

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

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