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

TL;DR

This paper investigates a variational problem involving a discontinuous weight function and a critical nonlinearity, establishing the existence of minimizers under certain conditions.

Contribution

It introduces a new analysis of a weighted variational problem with discontinuous weights and proves the existence of solutions in this setting.

Findings

Existence of minimizers for the weighted problem.

Analysis of the impact of discontinuous weights on solutions.

Conditions under which minimizers exist.

Abstract

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.