AbstractWe study the Dirichlet problem for the prescribed mean curvature equation in Minkowski space $$ \textstyle\begin{cases} \mathcal{M}(u)+ v^{\alpha }=0\quad \text{in } B, \\ \mathcal{M}(v)+ u^{\beta }=0\quad \text{in } B, \\ u_{\partial B}=v_{\partial B}=0, \end{cases} $$
{
M
(
u
)
+
v
α
=
0
in
B
,
M
(
v
)
+
u
β
=
0
in
B
,
u

∂
B
=
v

∂
B
=
0
,
where $\mathcal{M}(w)=\operatorname{div} ( \frac{\nabla w}{\sqrt{1\nabla w^{2}}} )$
M
(
w
)
=
div
(
∇
w
1
−

∇
w

2
)
and B is a unit ball in $\mathbb{R}^{N} (N\geq 2)$
R
N
(
N
≥
2
)
. We use the index theory of fixed points for completely continuous operators to obtain the existence, nonexistence and uniqueness results of positive radial solutions under some corresponding assumptions on α, β.