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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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Keywords
- Banach Space
- Continuous Operator
- Global Structure
- Global Behavior
- Auxiliary Equation
