Affiliation:
1. College of Information Science and Technology, Gansu Agricultural University , Lanzhou , P. R. China
Abstract
Abstract
In this article, we prove the existence of eigenvalues for the problem
(
ϕ
p
(
u
′
(
t
)
)
)
′
+
λ
h
(
t
)
ϕ
p
(
u
(
t
)
)
=
0
,
t
∈
(
0
,
1
)
,
A
u
(
0
)
−
A
′
u
′
(
0
)
=
0
,
B
u
(
1
)
+
B
′
u
′
(
1
)
=
0
\left\{\begin{array}{l}\left({\phi }_{p}\left(u^{\prime} \left(t)))^{\prime} +\lambda h\left(t){\phi }_{p}\left(u\left(t))=0,\hspace{1em}t\in \left(0,1),\\ Au\left(0)-A^{\prime} u^{\prime} \left(0)=0,\hspace{1em}Bu\left(1)+B^{\prime} u^{\prime} \left(1)=0\end{array}\right.
under hypotheses that
ϕ
p
(
s
)
=
∣
s
∣
p
−
2
s
,
p
>
1
{\phi }_{p}\left(s)={| s| }^{p-2}s,p\gt 1
, and
h
h
is a nonnegative measurable function on
(
0
,
1
)
\left(0,1)
, which may be singular at 0 and/or 1. For the result, we establish the existence of connected components of positive solutions for the following problem:
(
ϕ
p
(
u
′
(
t
)
)
)
′
+
λ
h
(
t
)
f
(
u
(
t
)
)
=
0
,
t
∈
(
0
,
1
)
,
u
(
0
)
=
0
,
a
u
′
(
1
)
+
c
(
λ
,
u
(
1
)
)
=
0
,
\left\{\begin{array}{l}\left({\phi }_{p}\left(u^{\prime} \left(t)))^{\prime} +\lambda h\left(t)f\left(u\left(t))=0,\hspace{1em}t\in \left(0,1),\\ u\left(0)=0,\hspace{1em}au^{\prime} \left(1)+c\left(\lambda ,u\left(1))=0,\end{array}\right.
where
λ
\lambda
is a real parameter,
a
≥
0
a\ge 0
,
f
∈
C
(
(
0
,
∞
)
,
(
0
,
∞
)
)
f\in C\left(\left(0,\infty ),\left(0,\infty ))
satisfies
inf
s
∈
(
0
,
∞
)
f
(
s
)
>
0
{\inf }_{s\in \left(0,\infty )}f\left(s)\gt 0
and
limsup
s
→
0
s
α
f
(
s
)
<
∞
{\mathrm{limsup}}_{s\to 0}{s}^{\alpha }f\left(s)\lt \infty
for some
α
>
0
\alpha \gt 0
.