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 $\kappa\in (-\infty, 0)\cup (0, \infty)$ 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). 
Ruyun Ma, Hongliang Gao, Tianlan Chen
math.CA