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.
Categories: math.AP