  Let $\Omega$ be a bounded domain in $\mathbb R^{N}$, $N\geq3$ with smooth
boundary,
  $a>0, \lambda>0$ and $0<\delta<3$ be real numbers. Define
$2^*:=\displaystyle\frac{2N}{N-2}$ and the characteristic function of a set $A$
by $\chi_A$.
  We consider the following critical problem with singular and discontinuous
nonlinearity: \begin{eqnarray*}
  (P_\la^a)~~~~ \qquad \Biggl\{\begin{array}{rl} -\Delta u &= \lambda
\left(u^{2^*-1}+ \displaystyle \chi_{\{u<a\}}u^{-\de} \right), u > 0~~\text{in}
~~\Omega, \\ u & = 0 ~\text{on}~
  \partial \Omega. \end{array} \end{eqnarray*}
  \noindent We study the existence and the global multiplicity of solutions to
the above problem.
