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

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 $\kappa \in (-\mathrm{\infty },0)\cup (0,\mathrm{\infty })$ 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).