  The problem
  \begin{equation}
  \label{bn}
  -\Delta u=|u|^{4\over n-2}u+\lambda V u\ \hbox{in}\ \Omega,\ u=0\ \hbox{on}\
\partial\Omega
  \end{equation}
  where $\Omega$ is a bounded regular domain in $\mathbb R^n$, $\lambda\in
\mathbb R$ and $V\in C^0(\overline \Omega),$ that was introduced by Brezis and
Nirenberg in their famous paper, where they address the existence of positive
solutions in the autonomous case, i.e. the potential $V$ is constant. Since
then, a huge amount of work has been done. In the following we will make a
brief history highlighting the results which are much closer to the problem we
wish to study in the present paper.
