The effects of a discontinues weight for a problem with a critical nonlinearity

Abstract

We study the minimizing problem $\inf \left\{{\int }_{\Omega }p(x)|\nabla u{|}^{2}dx,u\in H_0^1(\Omega ),\Vert u{\Vert }_{L^{\frac{2N}{N-2}}(\Omega )}=1\right\}$ where $\Omega$ is a smooth bounded domain of ${\mathbb{R}}^N$, $N\ge 3$ and $p$ a positive discontinuous function. We prove the existence of a minimizer under some assumptions.