# The Fredholm alternative for the $p$-Laplacian in exterior domains

## TL;DR

This paper studies the Fredholm alternative for the p-Laplacian operator in exterior domains, revealing solution multiplicity and differences from the entire space case using variational methods.

## Contribution

It establishes the Fredholm alternative for the p-Laplacian in exterior domains and demonstrates solution multiplicity with novel variational techniques.

## Key findings

- Proves the Fredholm alternative in exterior domains.
- Shows multiple solutions exist for the p-Laplacian.
- Discusses key differences from the entire space case.

## Abstract

We investigate the Fredholm alternative for the $p$-Laplacian in an exterior domain which is the complement of the closed unit ball in $\mathbb{R}^N$ ($N\geq 2$). By employing techniques of Calculus of Variations we obtain the multiplicity of solutions. The striking difference between our case and the entire space case is also discussed.