On a singular minimizing problem

TL;DR

This paper investigates a singular minimization problem involving a p-Laplacian with a singular term, establishing a new log-Sobolev inequality and identifying its optimal constant.

Contribution

It introduces a novel log-Sobolev inequality related to the singular p-Laplacian problem and determines the best constant for this inequality.

Findings

Established a new log-Sobolev type inequality.

Identified the best constant for the inequality.

Analyzed the minimization problem associated with the singular PDE.

Abstract

We study a minimizing problem associated with the singular problem \[ \left\{ \begin{array} [c]{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =\lambda u^{-1} & \mathrm{in\ }\Omega\\ u>0 & \mathrm{in\ }\Omega\\ u=0 & \mathrm{on\ }\partial\Omega, \end{array} \right. \] where $p>1$, $\lambda>0$ and $\Omega$ is a bounded and smooth domain of $\mathbb{R}^{N}$, $N\geq2.$ A new log-Sobolev type inequality is proved and the corresponding best constant is identifyied.