Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball

Abstract

We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation $$ \begin{cases} \displaystyle -{\rm div}\bigg( \frac{\nabla v} {\sqrt{1 - |\nabla v|^2}}\bigg)= f(|x|,v) &\quad \text{in } B_R, \\ v=0 & \quad \text{on } \partial B_R, \end{cases} $$ where $B_R$ is a ball in $\mathbb{R}^N$ ($N\ge 2$). According to the behaviour of $f=f(r,s)$ near $s=0$, we prove the existence of either one, two or three positive solutions. All results are obtained by reduction to an equivalent non-singular one-dimensional problem, to which variational methods can be applied in a standard way.