Non-standard eigenvalue problems for perturbed $p$-Laplacians

TL;DR

This paper investigates multi-parameter eigenvalue problems for perturbed p-Laplacians, introducing new non-standard eigenvalue problems and developing both classical and novel methods to analyze traveling wave solutions in nonlinear PDEs.

Contribution

It extends classical variational techniques to solve non-standard eigenvalue problems for perturbed p-Laplacians, proposing new methods for analyzing traveling waves.

Findings

Established dispersion relations between eigen-parameters

Proved existence of eigenvectors and positive eigenvectors

Developed analytical solutions for non-standard eigenvalue problems

Abstract

This paper is devoted to multi-parameter eigenvalue problems for perturbed $p$-Laplacians, modelling travelling waves for a class of non-linear evolution PDE. Dispersion relations between the eigen-para-meters, the existence of eigenvectors and positive eigenvectors, variational principles for eigenvalues of perturbed $p$-Laplacians and constructing analytical solutions are the main subject of this paper. Besides the $p$-Laplacian-like eigenvalue problems we also deal with new and non-standard eigenvalue problems, which can not be solved by the methods used in nonlinear eigenvalue problems for $p$-Laplacians and similar operators. We do both: extend and use classical variational and analytical techniques to solve standard eigenvalue problems and suggest new variational and analytical methods to solve the non-standard eigenvalue problems we encounter in the search for travelling waves.