Spectrum structure for eigenvalue problems involving mean curvature operators in Euclidean and Minkowski spaces

TL;DR

This paper investigates the spectral properties of a nonlinear eigenvalue problem involving mean curvature operators in Euclidean and Minkowski spaces, revealing finite zeros, symmetric humps, and the global solution structure.

Contribution

It provides a detailed analysis of the solution structure for a nonlinear eigenvalue problem with mean curvature operators, including zero distribution and symmetry properties.

Findings

Solutions have finitely many simple zeros.

All humps of solutions are identical.

The first hump is symmetric around the midpoint.

Abstract

In this paper, we are concerned with quasilinear Dirichlet problem $$ \left\{ \aligned &-\Big(\frac{u'(x)}{\sqrt{1+\kappa (u'(x))^2}}\Big)'=\lambda u(x), \ \ \ \ \ 0<x<1,\\ &u(0)= u(1)=0,\\ \endaligned \right. \eqno (P) $$ where $\kappa\in (-\infty, 0)\cup (0, \infty)$ is a constant. We show that any nontrivial solution $ u$ of (P) has only finite many of simple zeros in $[0,1]$, all of humps of $u$ are same, and the first hump is symmetric around the middle point of its domain. We also describe the global structure of the set of nontrivial solutions of (P).