  This paper addresses the following problem. \begin{equation}
  \left\{
  \begin{array}{lr}
  -{\Delta}u=\lambda I_\alpha*_\Omega u+|u|^{2^*-2}u\mbox{ in }\Omega
,\nonumber u\in H_0^1(\Omega).\nonumber
  \end{array}
  \right. \end{equation} Here, $\Omega$ is a bounded domain in $\mathbb{R}^N$
with $N\geq3$, $2^*=\frac{2N}{N-2}$, $\lambda\in\mathbb{R}$, $\lambda\in(0,N)$,
$I_\alpha$ is the Riesz potential and \begin{align} I_\alpha*_\Omega
u(x):=\int_\Omega
\frac{\Gamma(\frac{N-\alpha}{2})}{\Gamma(\frac{\alpha}{2})\pi^\frac{N}{2}2^\alpha|x-y|^{N-\alpha}}
u(y)dy. \nonumber \end{align} We study the non-existence, existence and
multiplicity results. Our argument combines Brezis-Nirenberg's method with the
regularity results involving potential terms. Especially, we study the
following nonlocal eigenvalue problem. \begin{equation}
  \left\{
  \begin{array}{lr}
  -{\Delta}u=\lambda I_\alpha*_\Omega u\mbox{ in }\Omega ,\nonumber
  \lambda\in\mathbb{R},\,u\in H_0^1(\Omega).\nonumber
  \end{array}
  \right. \end{equation}
