In this talk I will introduce a nonlocal version of the famous Br\'{e}zis-Nirenberg problem related to the critical Hartree equation. In fact we consider the minimizing problem

$$

S(\varepsilon V)=\displaystyle\inf\limits_{u\in H_{0}^{1}(\Omega)\backslash\{{0}\}}\ \ \frac{\int_{\Omega}|\nabla u|^{2}dx+\varepsilon\int_{\Omega}V| u|^{2}dx}{\Big(\int_{\Omega}\int_{\Omega}

\frac{|u(x)|^{2_{\mu}^{\ast}}|u(y)|^{2_{\mu}^{\ast}}}

{|x-y|^{\mu}}dxdy\Big)^{\frac{1}{2_{\mu}^{\ast}}}},

$$

where $\Omega$ is a bounded open subset of $\mathbb{R}^N$, $N\geq4$, $0<\mu<4$ and  $2_{\mu}^{\ast}=(2N-\mu)/(N-2)$ is the critical

exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under certain assumptions on $V$, we obtain the asymptotics of $S(\varepsilon V)-S(0)$ to leading order as $\varepsilon\rightarrow0^{+}$ and give a precise description of the blow-up profile of minimizing sequences. Moreover, we characterize the concentration points as being extrema of a quotient involving the Robin function. If $N=3$, we study the asymptotics of $S(Q+\varepsilon V)-S(Q)$ to leading order as $\varepsilon\rightarrow0^{+}$ for the following problem

$$

S(Q+\varepsilon V)=\displaystyle\inf\limits_{u\in H_{0}^{1}(\Omega)\backslash\{{0}\}}\ \ \frac{\int_{\Omega}|\nabla u|^{2}dx+\int_{\Omega}(Q+\varepsilon V)| u|^{2}dx}{\Big(\int_{\Omega}\int_{\Omega}

\frac{|u(x)|^{2_{\mu}^{\ast}}|u(y)|^{2_{\mu}^{\ast}}}

{|x-y|^{\mu}}dxdy\Big)^{\frac{1}{2_{\mu}^{\ast}}}},

$$

for $0<\mu<2$ and further give a precise description of the blow-up profile of minimizing sequences under certain assumptions on $Q$ and $V$. In particular, minimizers concentrate at a point in the zero set of the Robin function.