Positive radial solutions of a mean curvature equation in Minkowski space with strong singularity

Abstract:

The existence of positive radial solution is obtained for a mean curvature equation in Minkowski space of the form \[ \left \{ \begin {array}{ll} \hbox {div}(\frac {\nabla v}{\sqrt {1-|\nabla v|^2}})+f(|x|,v)=0\quad \textrm {in}\quad \Omega ; \\ v=0\quad \textrm {on}\quad \partial \Omega , \end {array} \right . \] where $\Omega$ is a unit ball in $\mathbb {R}^N$, $f(r,u)$ has singularities at $u=0$, $r=0$ and/or $r=1$. The main tool is the perturbation technique and nonlinear alternative of Leray-Schauder type. The interesting point is that the nonlinear term $f(r,u)$ at $u=0$ may be strongly singular.