Nonlinear eigenvalue problems and bifurcation for quasi-linear elliptic operators

TL;DR

This paper studies eigenvalue problems for quasi-linear elliptic operators, establishing regularity, simplicity of the first eigenvalue, and bifurcation phenomena, including multiple solutions, using variational and topological methods.

Contribution

It provides new insights into eigenfunctions' regularity, eigenvalue simplicity, and bifurcation analysis for quasi-linear elliptic operators, extending previous results with variational and topological techniques.

Findings

Eigenfunctions are in $L^{inite}$ and are $C^{1,eta}$ smooth.

The first eigenvalue is simple.

Existence of multiple critical points via variational methods.

Abstract

In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to $L^{\infty}$, which implies $C^{1,\alpha}$ smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray-Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski genus.