The first eigenvalue and eigenfunction of a nonlinear elliptic system
This is an incremental theoretical and numerical contribution for researchers studying eigenvalue problems in nonlinear elliptic systems.
The paper studies the first eigenvalue and eigenfunction of a nonlinear elliptic system with p-Laplacian, providing an alternative proof of simplicity, bounds, and a numerical algorithm that generates a decreasing sequence of positive numbers with numerical evidence of convergence.
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.