Author:
He Zhiqian,Zhao Yanzhong,Miao Liangying
Abstract
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.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory,Analysis