A note on the implicit function theorem for quasi-linear eigenvalue problems

TL;DR

This paper studies a quasi-linear eigenvalue problem involving the p-Laplacian and a Lipschitz continuous function, establishing existence, uniqueness, and continuous dependence of small solutions on the eigenvalue parameter.

Contribution

It proves the existence and uniqueness of small solutions for the problem without restrictions on the domain or the nonlinearity, and shows their continuous dependence on the eigenvalue.

Findings

Small solutions exist for small eigenvalues.

Solutions are unique among small solutions.

The solution curve depends continuously on the eigenvalue.

Abstract

We consider the quasi-linear eigenvalue problem $-\Delta_p u = \lambda g(u)$ subject to Dirichlet boundary conditions on a bounded open set $\Omega$, where $g$ is a locally Lipschitz continuous functions. Imposing no further conditions on $\Omega$ or $g$ we show that for small $\lambda$ the problem has a bounded solution which is unique in the class of all small solutions. Moreover, this curve of solutions depends continuously on $\lambda$.