Hindawi Publishing Corporation
Boundary Value Problems
Volume 2011, Article ID 715836, 12 pages
doi:10.1155/2011/715836
Research Article
Global Structure of Nodal Solutions for
Second-Order m-Point Boundary Value Problems
with Superlinear Nonlinearities
Yulian An
Department of Mathematics, Shanghai Institute of Technology, Shanghai 200235, China
Correspondence should be addressed to Yulian An, an
[email protected]
Received 8 May 2010; Revised 1 August 2010; Accepted 23 September 2010
Academic Editor: Feliz Manuel Minh
´
os
Copyright q 2011 Yulian An. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We consider the nonlinear eigenvalue problems u
λfu0, 0 <t<1, u00, u1
m−2
i1
α
i
uη
i
,wherem ≥ 3, η
i
0,t∈
0, 1
, 1.1
u
0
0,u
1
m−2
i1
α
i
u
η
i
.
1.2
Here m ≥ 3,η
i
∈ 0, 1, and α
i
to discuss the global structure of nodal solutions of 1.1, 1.2 with f
0
∞.
In this paper, we obtain a complete description of the global structure of nodal
solutions of 1.1, 1.2 under the following assumptions:
A1 α
i
> 0fori 1, ,m− 2, with 0 <
m−2
i1
α
i
< 1;
A2 f ∈ C
1
R \{0}, R ∩ CR, R satisfies fss>0fors
/
0;
A3 f
0
: lim
|s|→0
fs/s ∞;
A4 f
∞
: lim
|s|→∞
fs/s ∈ 0, ∞.
Let Y C0, 1 with the norm
1
m−2
i1
α
i
u
η
i
,
E
u ∈ C
2
0, 1
| u
0
0,u
1
∞
,
u
max
u
∞
,
u
∞
,
u
∞
ν
k
⊂ C
2
0, 1 consisting of functions
u ∈ C
2
0, 1 satisfying the following conditions:
S
ν
k
: i u00,νu
0 > 0,
ii u has only simple zeros in 0, 1 and has exactly k − 1zerosin0, 1;
T
ν
k
: i u00,νu
0 > 0andu
1
/
0,
ii u
has only simple zeros in 0, 1 and has exactly k zeros in 0, 1,
iii u has a zero strictly between each two consecutive zeros of u
Lu λu, u ∈ E. 1.7
We call the set of eigenvalues of 1.7 the spectrum of L and denote it by σL. The following
lemmas or similar results can be found in 1–3.
Lemma 1.3. Let A1 hold. The spectrum σL consists of a strictly increasing positive sequence of
eigenvalues λ
k
,k 1, 2, ,with corresponding eigenfunctions ϕ
k
xsin
λ
k
x. In addition,
i lim
k →∞
λ
k
∞;
ii ϕ
k
∈ T
k
, for each k ≥ 1, and ϕ
1
is strictly positive on 0, 1.
We can regard the inverse operator L
−1
: Y → E as an operator L
−1
k
. Let Σ
ν
k
denote the closure of set of those solutions of 1.1,
1.2 which belong to Φ
ν
k
. The main results of this paper are the following.
Theorem 1.5. Let (A1)–(A4) hold.
a If f
∞
0, then there exists a subcontinuum C
ν
k
of Σ
ν
k
with 0, 0 ∈C
ν
k
and
Proj
R
C
ν
k
0, ∞
. 1.9
c If f
∞
∞, then there exists a subcontinuum C
ν
k
of Σ
ν
k
with 0, 0 ∈C
ν
k
, Proj
R
C
ν
k
is a
bounded closed interval, and C
ν
k
approaches 0, ∞ as u→∞.
4 Boundary Value Problems
Theorem 1.6. Let (A1)–(A4) hold.
a If f
∞
0,then1.1, 1.2 has at least one solution in T
ν
k
for any λ ∈ 0, ∞.
f
n
0
−→ ∞ . 1.10
By means of the corresponding auxiliary equations, we obtain a sequence of unbounded
components {C
νn
k
} via Rabinowitz’s global bifurcation theorem 8, and this enables us to
find unbounded components C
ν
k
satisfying
0, 0
∈C
ν
k
⊂ lim sup C
νn
k
. 1.11
The rest of the paper is organized as follows. Section 2 contains some preliminary
propositions. In Section 3, we use the global bifurcation theorems to analyse the global
behavior of the components of nodal solutions of 1.1, 1.2.
2. Preliminaries
Definition 2.1 see 9.LetW be a Banach space and {C
Lemma 2.3 see 6. Assume that
i there exist z
n
∈ C
n
n 1, 2, and z
∗
∈ W, such that z
n
→ z
∗
;
ii r
n
∞,wherer
n
sup{x|x ∈ C
n
};
iii for all R>0,
∞
n1
C
n
∩ B
R
is a relative compact set of W,where
B
R
f
u
s
ds, 2.3
where
H
t, s
G
t, s
m−2
i1
α
i
G
η
i
,s
1 −
r
{
u ∈ Y |
u
∞
<r
}
. 2.5
Lemma 2.5. Let (A1)-(A2) hold. If u ∈ ∂Ω
r
,r>0,then
T
λ
u
∞
≤ λ
M
r
1
m−2
i1
α
}⊂Φ
ν
k
is a sequence of solutions of 1.1, 1.2.
Assume that μ
l
≤ C
0
for some constant C
0
> 0, and lim
l →∞
y
l
∞. Then
lim
l →∞
y
l
∞
∞. 2.7
Proof. From the relation y
l
tμ
l
m−2
i1
α
i
1 −
m−2
i1
α
i
η
i
1
0
f
y
l
s
ds, 2.8
f
s
,s∈
1
n
, ∞
∪
−∞, −
1
n
,
nf
1
n
s, s ∈
−
1
n
,
1
n
f
n
0
∞. 3.3
Now let us consider the auxiliary family of the equations
u
λf
n
u
0,t∈
0, 1
,
3.4
u
0
0,u
1
m−2
u ζ
n
u
nf
1
n
u ζ
n
u
. 3.6
Note that
lim
|
s
|
→ 0
ζ
n
s
s
0. 3.7
Let us consider
f
n
0
u
s
λζ
n
u
s
ds
: λL
−1
f
n
0
u
·
n
0
, 0 to
infinity in E. Moreover, {C
νn
k
}\{λ
k
/f
n
0
, 0}⊂Φ
ν
k
.
Proof of Theorem 1.5. Let us verify that {C
νn
k
} satisfies all of the conditions of Lemma 2.3.
Since
lim
n →∞
λ
k
f
n
∞, 3.11
and accordingly, ii holds. iii can be deduced directly from the Arzela-Ascoli Theorem and
the definition of f
n
. Therefore, the superior limit of {C
νn
k
}, D
ν
k
, contains an unbounded
connected component C
ν
k
with 0, 0 ∈C
ν
k
.
From the condition A2, applying Lemma 2.2 with p 2in10, we can show that the
initial value problem
v
λf
v
0,t∈
0, 1
ν
k
, we conclude that C
ν
k
⊂ Φ
ν
k
. Moreover,
C
ν
k
⊂ Σ
ν
k
by 1.1 and 1.2.
We divide the proof into three cases.
Case 1 f
∞
0. In this case, we show that Proj
R
C
ν
k
0, ∞.
Assume on the contrary that
sup
λ |
0
depending not on l. From Lemma 2.6, we have
lim
l →∞
y
l
∞
∞. 3.15
Set v
l
ty
l
t/y
l
∞
. Then v
l
∞
1. Now, choosing a subsequence and relabelling
if necessary, it follows that there exists μ
∗
,v
∗
∈ 0,C
f
y
l
t
y
l
∞
0. 3.18
The proof is similar to that of the step 1 of Theorem 1 in 7; we omit it. So, we obtain
v
∗
t
μ
∗
· 0 0,t∈
t ≡ 0fort ∈ 0, 1. This contradicts 3.16. Therefore
sup
λ |
λ, y
∈C
ν
k
∞. 3.21
Case 2 f
∞
∈ 0, ∞. In this case, we can show easily that C joins 0, 0 with λ
k
/f
∞
, ∞ by
using the same method used to prove Theorem 5.1in2.
Case 3 f
∞
∞. In this case, we show that C
ν
k
joins 0, 0 with 0, ∞.
Let {μ
l
,y
l
Taking subsequences again if necessary, we still denote {μ
l
,y
l
} such that {y
l
}⊂T
ν
k
∩ S
ν
k
.If
{y
l
}⊂T
ν
k
∩ S
ν
k1
, all the following proofs are similar.
Let
0 τ
0
l
<τ
1
l
< ···<τ
which is of length at least 1/k for some j ∈{0, 1, ,k − 1}. Then, for this
j, τ
j1
l
− τ
j
l
> 3/4k if l is large enough. Put τ
j
l
,τ
j1
l
I
j
l
.
Obviously, for the above given k, ν and j, y
l
t have the same sign on I
j
l
for all l.
Without loss of generality, we assume that
y
l
t
> 0,t∈ I
t
≥ inf
f
s
s
| 0 <s≤ M
1
> 0,t∈
τ
j
l
,τ
j1
l
, 3.27
and using the relation
y
l
t
μ
l
must change its sign on τ
j
l
,τ
j1
l
if l is large enough. This is a contradiction.
Hence {y
l
} is unbounded. From Lemma 2.6, we have that
lim
l →∞
y
l
∞
∞. 3.29
Note that {μ
l
,y
l
} satisfies the autonomous equation
y
l
μ
Armed with this information on the shape of y
l
, it is easy to show that for the above
given I
j
l
, {y
l
I
j
l
,∞
: max
I
j
l
y
l
t}
∞
l1
is an unbounded sequence. That is
lim
l →∞
y
l
τ
j
l
σ, τ
j1
l
− σ
. 3.32
This together with 3.31 implies that there exist constants α, β with α, β ⊂ I
j
∞
, such
that
lim
l →∞
y
l
t
∞, uniformly for t ∈
α, β
. 3.33
Hence, we have
lim
l →∞
f
lim
l →∞
μ
l
f
y
l
t
y
l
t
∞, uniformly for t ∈
α, β
. 3.35
From 3.28 we obtain that y
l
must change its sign on α, β if l is large enough. This is a
contradiction. Therefore lim
l →∞
μ
l
0.
λ
u
∞
≤ λ
M
r
1
m−2
i1
α
i
1 −
m−2
i1
α
i
η
i
1
0
G
s, s
η
i
1
0
G
s, s
ds r. 3.38
Then for λ ∈ 0,λ
r
and u ∈ ∂Ω
r
,
T
λ
u
∞
<
u
∞
. 3.39
This means that
Σ
k
.
Acknowledgments
The author is very grateful to the anonymous referees for their valuable suggestions. This
paper was supported by NSFC no.10671158, 11YZ225, YJ2009-16 no.A06/1020K096019.
References
1 B. P. Rynne, “Spectral properties and nodal solutions for second-order, m-point, boundary value
problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 12, pp. 3318–3327, 2007.
2 R. Ma and D. O’Regan, “Nodal solutions for second-order m-point boundary value problems with
nonlinearities across several eigenvalues,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64,
no. 7, pp. 1562–1577, 2006.
3 X. Xu, “Multiple sign-changing solutions for some m-point boundary-value problems,” Electronic
Journal of Differential Equations, vol. 2004, no. 89, pp. 1–14, 2004.
4 S. Jingxian, X. Xian, and D. O’Regan, “Nodal solutions for m-point boundary value problems using
bifurcation methods,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 10, pp. 3034–3046,
2008.
5 Y. An and R. Ma, “Global behavior of the components f or the second order m-point boundary value
problems,” Boundary Value Problems, vol. 2008, Article ID 254593, 10 pages, 2008.
6 R. Ma and Y. An, “Global structure of positive s olutions for superlinear second order m-point
boundary value problems,” Topological Methods in Nonlinear Analysis, vol. 34, no. 2, pp. 279–290, 2009.
7 R. Ma and B. Thompson, “Nodal solutions for nonlinear eigenvalue problems,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 59, no. 5, pp. 707–718, 2004.
12 Boundary Value Problems
8 P. H. Rabinowitz, “Some global results for nonlinear eigenvalue problems,” Journal of Functional
Analysis, vol. 7, pp. 487–513, 1971.
9 G. T. Whyburn, Topological Analysis, Princeton Mathematical Series. No. 23, Princeton University
Press, Princeton, NJ, USA, 1958.
10 Y. Naito and S. Tanaka, “Sharp conditions for the existence of sign-changing solutions to equations
involving the one-dimensional p-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol.
69, no. 9, pp. 3070–3083, 2008.
Copyright: Tài liệu đại học ©