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.
Geometric and topological approaches to shape variation in
Ginkgo
leaves
