On a singular minimizing problem

Abstract

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

Final version to appear in the Journal d'Analyse Mathématique

References in corpus (2)

• Asymptotics for the best Sobolev constants and their extremal functions
• Absolute continuity of the best Sobolev constant of a bounded domain

Cited by in corpus (1)

• Asymptotic behavior of extremals for fractional Sobolev inequalities associated with singular problems