Global structure of positive solutions for a Neumann problem with indefinite weight in Minkowski space

Abstract

We show the global structure of positive solutions for the Neumann problem involving mean curvature operator

$$\begin{aligned} \left\{ \begin{array}{ll} -(\frac{u'}{\sqrt{1-u'^2}})'=\lambda a(r)f(u), &{}\quad r\in (0,R), \\ u'(0)=u'(R)=0,&{} \\ \end{array} \right. \end{aligned}$$

(P)

where $\lambda >0$ is a parameter, $a:[0,R]\rightarrow {\mathbb {R}}$ is an $L^1$-function which is allowed to change sign and $f:[0,\infty )\rightarrow [0,\infty )$ is continuous. Depending on the behavior of f near 0 and $\infty$, we obtain that there exists $0<\lambda _*\le \lambda ^*$ such that for any $\lambda >\lambda ^*$, problem (P) possesses at least two positive solutions, while it has no solution for $\lambda \in (0,\lambda _*)$. The proof of the main results is based upon bifurcation method.