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

Abstract

In this paper, we are concerned with quasilinear Dirichlet problem $$ \left\{ \aligned &-\Big(\frac{u'(x)}{\sqrt{1+κ(u'(x))^2}}\Big)'=λu(x), \ \ \ \ \ 0<x<1,\\ &u(0)= u(1)=0,\\ \endaligned \right. \eqno (P) $$ 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).