The Green function for p-Laplace operators

TL;DR

This paper investigates the existence, uniqueness, and regularity of the Green function associated with a quasi-linear p-Laplace operator on bounded domains, extending classical potential theory to nonlinear operators.

Contribution

It establishes foundational results for the Green function of a nonlinear p-Laplace operator with a spectral parameter, including existence, uniqueness, and regularity properties.

Findings

Proves existence of the Green function for the nonlinear operator.

Demonstrates uniqueness under specified conditions.

Analyzes regularity properties of the Green function.

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 $-\Delta_p$.