Affiliation:
1. Lanzhou Petrochemical University of Vocational Technology , Lanzhou 730060 , P. R. China
2. Department of Mathematics, Northwest Normal University , Lanzhou 730070 , P. R. China
Abstract
Abstract
We study the one-parameter discrete Lane-Emden systems with Minkowski curvature operator
Δ
Δ
u
(
k
−
1
)
1
−
(
Δ
u
(
k
−
1
)
)
2
+
λ
μ
(
k
)
(
p
+
1
)
u
p
(
k
)
v
q
+
1
(
k
)
=
0
,
k
∈
[
2
,
n
−
1
]
Z
,
Δ
Δ
v
(
k
−
1
)
1
−
(
Δ
v
(
k
−
1
)
)
2
+
λ
μ
(
k
)
(
q
+
1
)
u
p
+
1
(
k
)
v
q
(
k
)
=
0
,
k
∈
[
2
,
n
−
1
]
Z
,
Δ
u
(
1
)
=
u
(
n
)
=
0
=
Δ
v
(
1
)
=
v
(
n
)
,
\left\{\begin{array}{ll}\Delta \left[\frac{\Delta u\left(k-1)}{\sqrt{1-{\left(\Delta u\left(k-1))}^{2}}}\right]+\lambda \mu \left(k)\left(p+1){u}^{p}\left(k){v}^{q+1}\left(k)=0,& k\in {\left[2,n-1]}_{{\mathbb{Z}}},\\ \Delta \left[\frac{\Delta v\left(k-1)}{\sqrt{1-{\left(\Delta v\left(k-1))}^{2}}}\right]+\lambda \mu \left(k)\left(q+1){u}^{p+1}\left(k){v}^{q}\left(k)=0,& k\in {\left[2,n-1]}_{{\mathbb{Z}}},\\ \Delta u\left(1)=u\left(n)=0=\Delta v\left(1)=v\left(n),& \\ \end{array}\right.
where
n
∈
N
n\in {\mathbb{N}}
with
n
>
4
n\gt 4
,
max
{
p
,
q
}
>
1
\max \left\{p,q\right\}\gt 1
,
λ
>
0
\lambda \gt 0
,
Δ
u
(
k
−
1
)
=
u
(
k
)
−
u
(
k
−
1
)
\Delta u\left(k-1)=u\left(k)-u\left(k-1)
, and
μ
(
k
)
>
0
\mu \left(k)\gt 0
for all
k
∈
[
2
,
n
−
1
]
Z
k\in {\left[2,n-1]}_{{\mathbb{Z}}}
. The existence of zero at least one or two positive solutions for the system are obtained according to the different intervals of
λ
\lambda
. Our main tools are based on topological methods, critical point theory, and lower and upper solutions.