Abstract
AbstractWe consider the existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear difference equation $$ \textstyle\begin{cases} -\nabla [\phi (\triangle u(t))]=\lambda a(t,u(t))+\mu b(t,u(t)), \quad t\in \mathbb{T}, \\ u(1)=u(N)=0, \end{cases} $$
{
−
∇
[
ϕ
(
△
u
(
t
)
)
]
=
λ
a
(
t
,
u
(
t
)
)
+
μ
b
(
t
,
u
(
t
)
)
,
t
∈
T
,
u
(
1
)
=
u
(
N
)
=
0
,
where $\lambda ,\mu \geq 0$
λ
,
μ
≥
0
, $\mathbb{T}=\{2,\ldots ,N-1\}$
T
=
{
2
,
…
,
N
−
1
}
with $N>3$
N
>
3
, $\phi (s)=s/\sqrt{1-s^{2}}$
ϕ
(
s
)
=
s
/
1
−
s
2
. The function $f:=\lambda a(t,s)+\mu b(t,s)$
f
:
=
λ
a
(
t
,
s
)
+
μ
b
(
t
,
s
)
is either sublinear, or superlinear, or sub-superlinear near $s=0$
s
=
0
. Applying the topological method, we prove the existence of either one or two, or three positive solutions.
Funder
National Natural Science Foundation of China
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Algebra and Number Theory,Analysis
Cited by
2 articles.
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