# The first eigenvalue and eigenfunction of a nonlinear elliptic system

## TL;DR

This paper investigates the principal eigenvalue and eigenfunction of a nonlinear elliptic system involving the p-Laplacian, providing analytical proofs, bounds, and a convergent numerical algorithm for approximation.

## Contribution

It offers a new analytical proof of eigenvalue simplicity, establishes bounds, and develops a numerical method for approximating the first eigenvalue of nonlinear elliptic systems.

## Key findings

- Proved the simplicity of the first eigenvalue.
- Derived upper and lower bounds for the eigenvalue.
- Developed a convergent numerical algorithm.

## Abstract

In this paper, we study the first eigenvalue of a nonlinear elliptic system involving $p$-Laplacian as the differential operator. The principal eigenvalue of the system and the corresponding eigenfunction are investigated both analytically and numerically. An alternative proof to show the simplicity of the first eigenvalue is given. In addition, the upper and lower bounds of the first eigenvalue are provided. Then, a numerical algorithm is developed to approximate the principal eigenvalue. This algorithm generates a decreasing sequence of positive numbers and various examples numerically indicate its convergence. Further, the algorithm is generalized to a class of gradient quasilinear elliptic systems.