Generalized elliptic functions and their application to a nonlinear eigenvalue problem with $p$-Laplacian

TL;DR

This paper generalizes elliptic functions to analyze a nonlinear eigenvalue problem involving the p-Laplacian, providing explicit spectral descriptions and revealing connections to related eigenvalue problems.

Contribution

It introduces a generalized form of elliptic functions and applies them to solve and describe the spectra of p-Laplacian eigenvalue problems in closed form.

Findings

Complete spectral description of the eigenvalue problem.

Explicit eigenfunction representations in terms of parameters.

Identification of solutions related to p/2-Laplacian eigenvalue problems.

Abstract

The Jacobian elliptic functions are generalized and applied to a nonlinear eigenvalue problem with $p$-Laplacian. The eigenvalue and the corresponding eigenfunction are represented in terms of common parameters, and a complete description of the spectra and a closed form representation of the corresponding eigenfunctions are obtained. As a by-product of the representation, it turns out that a kind of solution is also a solution of another eigenvalue problem with $p/2$-Laplacian.