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Multiple Nodal Solutions for Some Fourth-Order Boundary Value Problems via Admissible Invariant Sets
Boundary Value Problems volume 2008, Article number: 403761 (2008)
Abstract
Existence and multiplicity results for nodal solutions are obtained for the fourth-order boundary value problem (BVP) ,
,
, where
is continuous. The critical point theory and admissible invariant sets are employed to discuss this problem.
1. Introduction
In this paper, we consider the existence of nodal solutions to the semilinear fourth-order equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ2_HTML.gif)
where is continuous.
Owning to the importance of higher-order differential equations in physics, the existence and multiplicity of the solutions to such problems have been studied by many authors. They obtained the existence of solutions by the cone expansion or compression fixed point theorem [1–6]; sub-sup solution method [7–9]; critical point theory [10–13]; Morse theory [14, 15]; and eta [16, 17]. There are also papers which study nodal solutions for elliptic equations [18, 19]. In particular, in [20], Han and Li obtained multiple positive, negative, and sign-changing solutions by combining the critical point theory and the method of sub-sup solutions for the (BVP) (1.2). The main result is as follows:
there exist a strict subsolution
and a strict supersolution
of (BVP) (1.2) with
,
, and
;
is strictly increasing in
;
is locally Lipschitz continuous in
;
there exist
and
such that
for all
and
Theorem 1.1 (see [20]).
Assume that hold. Then, (BVP) (1.2) has at least four solutions.
Motivated by their ideas, we cannot help wondering if there are no strict subsolution and supersolution of (BVP) (1.2), can we still get the nodal solutions just by critical point theory? In this paper, we will use the admissible invariant sets and critical point theory to settle this problem. But we should point out that in all theorems of our paper, the nonlinearity is assumed to be odd in
, while no such symmetry is required in [20].
The paper is organized as follows: in Section 2, we give some preliminaries, including the critical point theorems which will be used in our main results and some concepts concerning the partially ordered Banach space. The main results and proofs are established in Section 3.
2. Preliminaries
Let be a Hilbert space and
a Banach space densely embedded in
. Assume that
has a closed convex cone
and that
has interior points in
, that is,
with
the interior and
the boundary of
in
.
Let and
for
. We use the following notation:
,
,
,
for
. Let
and
denote the norms in
and
, respectively.
Lemma 2.1 (see [21]).
Assume is a Hilbert space, and
is a closed convex set of
,
, and
. Then, there exists a pseudogradient vector field
for
, and
. Furthermore, if
is even,
, then
is odd.
Consider the pseudogradient flow on
associated with the vector field
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ3_HTML.gif)
We see that is odd in
, if
is odd in
. Since
for
and
the Brezis-Martin theorem [22] implies that
for
.
Definition 2.2 (see [21, 23]).
With the flow , a subset
is called an invariant set if
for
.
Let us assume that
(F),
for
,
is continuous.
Under condition , we have
for
and
is continuous in
.
Definition 2.3 (see [21]).
Let be an invariant set under
.
is said to be an admissible invariant set for
if (a)
is the closure of an open set in
, that is,
; (b) if
for some
and
in
as
for some
, then
in
; (c) if
such that
in
, then
in
; (d) for any
,
for
.
Lemma 2.4 (see [24]).
Let and
hold. Assume
is even, bounded from below,
and satisfies (PS) condition. Assume that the positive cone
is an admissible invariant set for
and
for all
. Suppose there is a linear subspace
with
, such that
for some
, where
Then,
has at least
pairs of critical points with negative critical values. More precisely,
(i)if,
has at least one pair of critical points in
and at least
pairs of critical points in
(ii)ifhas at least one pair of critical points in
and at least
pairs of critical points in
Lemma 2.5 (see [21]).
Let and
hold. Assume
is even,
, and
satisfies (PS) condition. Assume that the positive cone
is an admissible invariant set for
and
for all
. Suppose there exist linear subspaces
and
with
,
(
, resp.),
, such that for some
,
and
. Then,
has at least
(
, resp.) pairs of critical points in
with negative critical values.
Lemma 2.6 (see [21]).
Let and
hold. Assume
is even,
and
satisfies (PS) condition. Assume that the positive cone
is an admissible invariant set for
and
for all
. Suppose there exist linear subspaces
and
with
,
,
, such that for some
,
and
. Then for
(
, resp.),
has at least
(
, resp.) pairs of critical points in
with positive critical values.
Assume is even,
, satisfies
and
condition for
. Assume that
is an admissible invariant set for
,
for all
.
, where
are finite-dimensional subspaces of
, and for each
, let
and
Assume for each
there exist
such that
, where
,
as
. Then,
has a sequence of critical points
such that
as
, provided
for large
.
Next, we need some basic concepts of ordered Banach spaces.
Definition 2.8.
An ordered real Banach space is a pair , where
is a real Banach space and
a closed convex subset of
such that
and
. The partial order on
is given by the cone
. For
, we write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ4_HTML.gif)
If has nonempty interior, then it is called a solid cone. If every ordered interval is bounded, then
is called a normal cone. An operator
is called order preserving (in the literature sometimes increasing) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ5_HTML.gif)
strictly order preserving if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ6_HTML.gif)
and strongly order preserving if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ7_HTML.gif)
3. Main Results
In this section, we will employ the abstract results in Section 2 to establish some existence theorems on sign-changing solutions of (BVP) (1.2). Firstly, we give some lemmas to change (BVP) (1.2) to a variational problem. Let be the usual real Banach space with the norm
for all
. We can easily verify that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ8_HTML.gif)
is also a Banach space with respect to . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ9_HTML.gif)
then is a normal solid cone in
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ10_HTML.gif)
By , we denote the usual real Hilbert space with the inner product
for all
It is well known that the solution of (BVP) (1.2) in is equivalent to the solution of the following integral equation in
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ11_HTML.gif)
where is the Green's function of the linear boundary value problem
for all
subject to
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ12_HTML.gif)
Define operators by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ13_HTML.gif)
Since , (3.4) is equivalent to the following operator equation in
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ14_HTML.gif)
Remark 3.1.
It is easy to see that
-
(i)
is nonnegative continuous;
-
(ii)
;
-
(iii)
is bounded and continuous.
Lemma 3.2 (see [20]).
is a linear completely continuous operator and also a linear completely continuous operator from
In addition,
is strongly order-preserving.
From the definition of , we can obtain that
for all
with
. Therefore,
for all
with
. It is well known that all eigenvalues of
are
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ15_HTML.gif)
which have the corresponding orthonormal eigenfunctions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ16_HTML.gif)
and .
Lemma 3.3 (see [10]).
The operator equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ17_HTML.gif)
has a solution in if and only if the operator equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ18_HTML.gif)
has a solution in .
The uniqueness of the solution for these two above equations is also equivalent.
Remark 3.4.
From the proof of Lemma 3.3 [10], it is very clear if is a solution for (3.11), then
is a solution for (3.7). Furthermore, if
is a solution for (3.11), then
is a solution for (3.7) with the same sign, which follows from Lemma 3.2.
Lemma 3.5 (see [10]).
Let ,
. Then,
(i)is Fréchet differentiable on
and
for all
(ii)is Fréchet differentiable on
and
for all
Choose and
to be our Hilbert space and Banach space, respectively. Define a functional
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ19_HTML.gif)
Then, according to Lemma 3.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ20_HTML.gif)
Hence, Lemma 3.3 implies that the operator equation has a solution in
if and only if the functional
has a critical point in
. Thus, (BVP) (1.2) has been transformed into a variational problem.
We refer the following assumption:
is continuous and increasing in
.
Lemma 3.6.
Under ,
is satisfied, and
is strongly order-preserving.
Proof.
The proof is similar to [20], and we omit it here.
Lemma 3.7.
Under ,
is an admissible invariant set for
.
Proof.
We know that is strongly order-preserving, so does
given in Lemma 2.1. The Brezis-Martin theory implies that
and
are invariant sets under the negative pesudogradient flow of
. Requirement (a) is satisfied automatically. For (d), we note that for all
, we have
, similar to the proof in [23],
. To prove (b), let
for some
, so
, let
be a sequence such that
in
for some
, then
in
. For (c), if
, then
, if
in
, for
, then
and
, so
in
, and the proof is completed.
Lemma 3.8 (see [15]).
Any bounded sequence such that
as
has a convergent subsequence.
Next, we make more assumptions:
(f
2
) uniformly for
;
(f
3
), uniformly for
and some
;
(f
4
) is odd in
.
Theorem 3.9 (sublinear nonlinearity).
Under , (BVP) (1.2) has at least one pair of one-sign solutions
,
, and at least
pairs of nodal solutions
for
.
Proof.
It is easy to see that and
holds.
is an admissible invariant set for
, and
for
. Also,
is even,
. By
, there exist
,
such that
for all
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ21_HTML.gif)
So is coercive, bounded from below, and satisfies (PS) condition.
Take ; from
, there exist
,
such that
,
, choose
, then
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ22_HTML.gif)
so for
small. Result follows from Lemma 2.4.
Next, we consider an asymptotically linear problem:
(f
5
) uniformly for
;
(f
6
), uniformly for
.
Theorem 3.10 (asymptotically linear case).
Under ,
,
, and
, (BVP) (1.2) has at least
pairs of nodal solutions provided
or
. Here,
, if
; and
, if
.
Proof.
Take and
such that for
,
. Now let
be a (PS) sequence for
. Writing
with
,
, and taking inner product of
and
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ23_HTML.gif)
So is bounded, where
. Then,
satisfies the (PS) condition.
If , let
, and
, then
, and
.
From , we know that there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ24_HTML.gif)
Then, for ,
, we can obtain, when
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ25_HTML.gif)
So, choose , then
.
From , we can get there exist
,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ26_HTML.gif)
Then, when , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ27_HTML.gif)
Choose large enough such that
, and
, result follows from Lemma 2.6.
If , let
,
, then
,
. From (3.17), when
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ28_HTML.gif)
When , we know from (3.19),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ29_HTML.gif)
which means , then result follows from Lemma 2.5.
Next, we consider a superlinear problem. Assume that
there is
such that
for
large;
there are
,
such that
for
large.
Theorem 3.11 (superlinear nonlinearity).
Under ,
,
, and
, (BVP) (1.2) has infinitely many nodal solutions.
Proof.
From condition by the standard argument,
satisfies
condition for every
. Let
. From
, we obtain
for all
. Define
, it is very clear
and
, so
and
. So if
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ30_HTML.gif)
Choosing , we obtain, if
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ31_HTML.gif)
Let . From
, after integrating, we obtain the existence of
such that
for
. Hence, we have
for
and
is constant. Therefore, when
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ32_HTML.gif)
Noting , choose
large enough, such that
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F403761/MediaObjects/13661_2008_Article_803_Equ33_HTML.gif)
Result follows from Lemma 2.7.
Remark 3.12.
If there exist no strict supsolution and supersolution required in [20], just only using the functional to get the critical point [10, 11], then we just know that (BVP) (1.2) has solutions, even we can know the sign of the critical point of the functional
because
is not strongly order-preserving in
. In our paper, using admissible invariant sets in
, we can settle the problem.
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Acknowledgments
The authors are grateful to the referees for their useful suggestions which have improved the writing of the paper. Jihui Zhang thanks Z. Zhang and the members of AMSS very much for their hospitality and invitation to visit the Academy of Mathematics and Systems Sciences (AMSS), Academia Sinica, in January 2008. The authors also would like to thank Professor D. Cao, Professor S. Li, Professor Y. Ding, and Professor H. Yin for their help and many valuable discussions. This research was supported by the NNSF of China (Grant no.10871096), Foundation of Major Project of Science and Technology of Chinese Education Ministry, SRFDP of Higher Education, and NSF of Education Committee of Jiangsu Province. Zhitao Zhang was supported by NNSF of China (Grant no.10671195).
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Yang, Y., Zhang, J. & Zhang, Z. Multiple Nodal Solutions for Some Fourth-Order Boundary Value Problems via Admissible Invariant Sets. Bound Value Probl 2008, 403761 (2008). https://doi.org/10.1155/2008/403761
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DOI: https://doi.org/10.1155/2008/403761