The Green function for p-Laplace operators

Abstract

On a bounded domain $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$, we consider existence, uniqueness and "regularity" issues for the Green function $G_{\lambda }$ of the quasi-linear operator $u\to -{\Delta }_{p}u-\lambda |u{|}^{p-2}u$ with $1<p\le N$, homogeneous Dirichlet boundary condition and $\lambda <{\lambda }_1$, where ${\lambda }_1>0$ is the first eigenvalue of $-{\Delta }_p$.