Positive solutions for Dirichlet problems involving the mean curvature operator in Minkowski space

Abstract

We prove the existence of classical positive radial solutions to the boundary value problems

$$\begin{aligned} \left\{ \begin{array}{ll} -\,\text {div}\big (\frac{\nabla y}{\sqrt{1-|\nabla y|^2}}\big )=\lambda a(|x|)f(y) \ \ \ \ \ &{} \text {in}\ B(b),\\ y=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \; &{} \text {on}\ \partial B(b),\\ \end{array} \right\} . \end{aligned}$$

where $b>0$, $B(b)=\{x\in \mathbb {R}^N:|x|<b\}$, $a:[0, b]\rightarrow \mathbb {R}$ is a continuous function which may change sign, $f:[0,\infty )\rightarrow \mathbb {R}$ is a continuous function with $f(s)>0$ in [0, b], and $\lambda >0$ is sufficiently small. Our approach is based on the Leray–Schauder fixed point theorem.