  We study the minimizing problem
$\inf\left\{\displaystyle\int_{\Omega}p(x)|\nabla u|^{2}dx,\,u\in
H^{1}_{0}(\Omega),\,\|u\|_{L^{\frac{2N}{N-2}}(\Omega)}=1\right\}$ where
$\Omega$ is a smooth bounded domain of $\R^{N}$, $N\geq 3$ and $p$ a positive
discontinuous function. We prove the existence of a minimizer under some
assumptions.
