The Green function for p-Laplace operators

Abstract

On a bounded domain $\Omega \subset \mathbb{R}^N$, $N\geq 2$, 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 $1<p \leq N$, homogeneous Dirichlet boundary condition and $\lambda<\lambda_1$, where $\lambda_1>0$ is the first eigenvalue of $-\Delta_p$.