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Global Structure of Nodal Solutions for Second-Order m-Point Boundary Value Problems with Superlinear Nonlinearities
Boundary Value Problems volume 2011, Article number: 715836 (2011)
Abstract
We consider the nonlinear eigenvalue problems ,
,
,
, where
,
and
for
with
and
satisfies
for
, and
, where
. We investigate the global structure of nodal solutions by using the Rabinowitz's global bifurcation theorem.
1. Introduction
We study the global structure of nodal solutions of the problem


Here and
for
with
is a positive parameter, and
In the case that the global structure of nodal solutions of nonlinear second-order
-point eigenvalue problems (1.1), (1.2) have been extensively studied; see [1–5] and the references therein. However, relatively little is known about the global structure of solutions in the case that
and few global results were found in the available literature when
The likely reason is that the global bifurcation techniques cannot be used directly in the case. On the other hand, when
-point boundary value condition (1.2) is concerned, the discussion is more difficult since the problem is nonsymmetric and the corresponding operator is disconjugate. In [6], we discussed the global structure of positive solutions of (1.1), (1.2) with
However, to the best of our knowledge, there is no paper to discuss the global structure of nodal solutions of (1.1), (1.2) with
In this paper, we obtain a complete description of the global structure of nodal solutions of (1.1), (1.2) under the following assumptions:
for
with
satisfies
for
;
.
Let with the norm

Let

with the norm

respectively. Define by setting

Then has a bounded inverse
and the restriction of
to
that is,
is a compact and continuous operator; see [1, 2, 6].
For any function
if
then
is a simple zero of
if
For any integer
and any
define sets
consisting of functions
satisfying the following conditions:
(i)
(ii) has only simple zeros in
and has exactly
zeros in
;
(i) and
(ii) has only simple zeros in
and has exactly
zeros in
(iii) has a zero strictly between each two consecutive zeros of
.
Remark 1.1.
Obviously, if then
or
The sets
are open in
and disjoint.
Remark 1.2.
The nodal properties of solutions of nonlinear Sturm-Liouville problems with separated boundary conditions are usually described in terms of sets similar to see [7]. However, Rynne [1] stated that
are more appropriate than
when the multipoint boundary condition (1.2) is considered.
Next, we consider the eigenvalues of the linear problem

We call the set of eigenvalues of (1.7) the spectrum of and denote it by
. The following lemmas or similar results can be found in [1–3].
Lemma 1.3.
Let hold. The spectrum
consists of a strictly increasing positive sequence of eigenvalues
with corresponding eigenfunctions
In addition,
(i)
(ii) for each
and
is strictly positive on
We can regard the inverse operator as an operator
In this setting, each
is a characteristic value of
with algebraic multiplicity defined to be dim
where
denotes null-space and
is the identity on
Lemma 1.4.
Let hold. For each
the algebraic multiplicity of the characteristic value
of
is equal to 1.
Let under the product topology. As in [7], we add the points
to our space
Let
Let
denote the closure of set of those solutions of (1.1), (1.2) which belong to
The main results of this paper are the following.
Theorem 1.5.
Let (A1)–(A4) hold.
(a)If then there exists a subcontinuum
of
with
and

(b)If then there exists a subcontinuum
of
with

(c)If then there exists a subcontinuum
of
with
is a bounded closed interval, and
approaches
as
Theorem 1.6.
Let (A1)–(A4) hold.
(a)If , then (1.1), (1.2) has at least one solution in
for any
(b)If , then (1.1), (1.2) has at least one solution in
for any
(c)If then there exists
such that (1.1), (1.2) has at least two solutions in
for any
.
We will develop a bifurcation approach to treat the case . Crucial to this approach is to construct a sequence of functions
which is asymptotic linear at
and satisfies

By means of the corresponding auxiliary equations, we obtain a sequence of unbounded components via Rabinowitz's global bifurcation theorem [8], and this enables us to find unbounded components
satisfying

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]).
Let be a Banach space and
a family of subsets of
Then the superior limit
of
is defined by

Lemma 2.2 (see [9]).
Each connected subset of metric space is contained in a component, and each connected component of
is closed.
Lemma 2.3 (see [6]).
Assume that
(i)there exist and
such that
;
(ii), where
;
(iii)for all is a relative compact set of
, where

Then there exists an unbounded connected component in
and
.
Define the map by

where

It is easy to verify that the following lemma holds.
Lemma 2.4.
Assume that (A1)-(A2) hold. Then is completely continuous.
For , let

Lemma 2.5.
Let (A1)-(A2) hold. If , then

where
Proof.
The proof is similar to that of Lemma in [6]; we omit it.
Lemma 2.6.
Let (A1)-(A2) hold, and is a sequence of solutions of (1.1), (1.2). Assume that
for some constant
, and
Then

Proof.
From the relation we conclude that
Then

which implies that is bounded whenever
is bounded.
3. Proof of the Main Results
For each define
by

Then with

By (A3), it follows that

Now let us consider the auxiliary family of the equations


Lemma 3.1 (see [1, Proposition ]).
Let (A1), (A2) hold. If is a nontrivial solution of (3.4), (3.5), then
for some
.
Let be such that

Note that

Let us consider

as a bifurcation problem from the trivial solution
Equation (3.8) can be converted to the equivalent equation

Further we note that for
near
in
The results of Rabinowitz [8] for (3.8) can be stated as follows. For each integer , there exists a continuum
of solutions of (3.8) joining
to infinity in
Moreover,
Proof of Theorem 1.5.
Let us verify that satisfies all of the conditions of Lemma 2.3.
Since

condition in Lemma 2.3 is satisfied with
. Obviously

and accordingly, holds.
can be deduced directly from the Arzela-Ascoli Theorem and the definition of
. Therefore, the superior limit of
, contains an unbounded connected component
with
.
From the condition (A2), applying Lemma 2.2 with in [10], we can show that the initial value problem

has a unique solution on for every
and
. Therefore, any nontrivial solution
of (1.1), (1.2) has only simple zeros in
and
. Meanwhile, (A1) implies that
[1, proposition 4.1]. Since
, we conclude that
. Moreover,
by (1.1) and (1.2).
We divide the proof into three cases.
Case 1 ().
In this case, we show that .
Assume on the contrary that

then there exists a sequence such that

for some positive constant depending not on
. From Lemma 2.6, we have

Set Then
Now, choosing a subsequence and relabelling if necessary, it follows that there exists
with

such that

Since , we can show that

The proof is similar to that of the step 1 of Theorem in [7]; we omit it. So, we obtain


and subsequently, for
. This contradicts (3.16). Therefore

Case 2 ().
In this case, we can show easily that joins
with
by using the same method used to prove Theorem
in [2].
Case 3 ().
In this case, we show that joins
with
.
Let be such that

If is bounded, say,
, for some
depending not on
, then we may assume that

Taking subsequences again if necessary, we still denote such that
. If
, all the following proofs are similar.
Let

denote the zeros of in
. Then, after taking a subsequence if necessary,
. Clearly,
. Set
. We can choose at least one subinterval
which is of length at least
for some
. Then, for this
if
is large enough. Put
.
Obviously, for the above given and
have the same sign on
for all
. Without loss of generality, we assume that

Moreover, we have

Combining this with the fact

and using the relation

we deduce that must change its sign on
if
is large enough. This is a contradiction. Hence
is unbounded. From Lemma 2.6, we have that

Note that satisfies the autonomous equation

We see that consists of a sequence of positive and negative bumps, together with a truncated bump at the right end of the interval
with the following properties (ignoring the truncated bump) (see, [1]):
(i)all the positive (resp., negative) bumps have the same shape (the shapes of the positive and negative bumps may be different);
(ii)each bump contains a single zero of , and there is exactly one zero of
between consecutive zeros of
;
(iii)all the positive (negative) bumps attain the same maximum (minimum) value.
Armed with this information on the shape of it is easy to show that for the above given
is an unbounded sequence. That is

Since is concave on
, for any
small enough,

This together with (3.31) implies that there exist constants with
, such that

Hence, we have

Now, we show that .
Suppose on the contrary that, choosing a subsequence and relabeling if necessary, for some constant
. This implies that

From (3.28) we obtain that must change its sign on
if
is large enough. This is a contradiction. Therefore
.
Proof of Theorem 1.6.
and
are immediate consequence of Theorem 1.5
and
, respectively.
To prove , we rewrite (1.1), (1.2) to

By Lemma 2.5, for every and
,

where
Let be such that

Then for and
,

This means that

By Lemma 2.6 and Theorem 1.5, it follows that is also an unbounded component joining
and
in
. Thus, (3.40) implies that for
(1.1), (1.2) has at least two solutions in
.
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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).
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An, Y. Global Structure of Nodal Solutions for Second-Order m-Point Boundary Value Problems with Superlinear Nonlinearities. Bound Value Probl 2011, 715836 (2011). https://doi.org/10.1155/2011/715836
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DOI: https://doi.org/10.1155/2011/715836