- Research Article
- Open access
- Published:
Nontrivial Solutions of the Asymmetric Beam System with Jumping Nonlinear Terms
Boundary Value Problems volume 2010, Article number: 728101 (2010)
Abstract
We investigate the existence of multiple nontrivial solutions for perturbations
and
of the beam system with Dirichlet boundary condition
in
,
in
, where
, and
are nonzero constants. Here
is the beam operator in
, and the nonlinearity
crosses the eigenvalues of the beam operator.
1. Introduction
Let be the beam operator in
,
In this paper, we investigate the existence of multiple nontrivial solutions
for perturbations
of the beam system with Dirichlet boundary condition

where and the nonlinearity
crosses the eigenvalues of the beam operator. This system represents a bending beam supported by cables in the two directions.
In [1, 2], the authors investigated the multiplicity of solutions of a nonlinear suspension bridge equation in an interval

where the nonlinearity crosses an eigenvalue. This equation represents a bending beam supported by cables under a load
The constant
represents the restoring force if the cables stretch. The nonlinearity
models the fact that cables resist expansion but do not resist compression.
In [2] Lazer and McKenna point out that the kind of nonlinearity

can furnish a model to study travelling waves in suspension bridges. This is a one-dimensional beam equation that represents only the up and down travelling waves of the beam. But the beam has also the right and left travelling waves. Hence we can consider two-dimensional beam equation (1.1).
The nonlinear equation with jumping nonlinearity has been extensively studied by many authors. For the fourth order elliptic equation, Taratello [3] and Micheletti and Pistoia [4, 5] proved the existence of nontrivial solutions, by using degree theory and critical point theory, separately. For one-dimensional case, Lazer and McKenna [6] proved the existence of nontrivial solution by the global bifurcation method. For this jumping nonlinearity, we are interested in the multiple nontrivial solutions of the equation. Here we used variational reduction method to find the nontrivial solutions of problem (1.1).
In Section 2, we investigate some properties of the Hilbert space spanned by eigenfunctions of the beam operator. We show that only the trivial solution exists for problem (1.4) when , and
. In Section 3, we state the Mountain Pass Theorem. In Section 4, we investigate the existence of nontrivial solutions
for a perturbation
of the asymmetric beam equation

where , and
are constants. This equation satisfies Dirichlet boundary condition on the interval
and periodic condition on the variable
. We use the variational reduction method to apply mountain pass theorem in order to get the main result that for
(1.2) has at least three periodic solutions, two of which are nontrivial. In Section 5, we investigate the existence of multiple nontrivial solutions
for perturbations
of beam system (1.1). We also prove that for
(1.1) has only the trivial solution.
2. Preliminaries
Let be the differential operator and
Then the eigenvalue problem

has infinitely many eigenvalues and corresponding normalized eigenfunctions
given by

We note that all eigenvalues in the interval are given by

Let be the square
and
the Hilbert space defined by

Then the set of functions is an orthonormal basis in
. Let us denote an element
in
as

and we define a subspace of
as

Then this is a complete normed space with a norm

Since for all
, we have that
(), where
denotes the
norm of
;
() if and only if
.
Define . Then we have the following lemma (cf. [7]).
Lemma 2.1.
Let ,
. Then we have that

Theorem 2.2.
Let , and
. Then the equation, with Dirichlet boundary condition,

has only the trivial solution in .
Proof.
Since and
, let
. The equation is equivalent to

By Lemma 2.1, is a compact linear map from
into
. Therefore, it is
norm
. We note that

So the right-hand side of (2.10) defines a Lipschitz mapping of into
with Lipschitz constant
. Therefore, by the contraction mapping principle, there exists a unique solution
. Since
is a solution of (2.10),
is the unique solution.
3. Mountain Pass Theorem
The mountain pass theorem concerns itself with proving the existence of critical points of functional which satisfy the Palais-Smale (PS) condition, which occurs repeatedly in critical point theory.
Definition 3.1.
We say that satisfies the Palais-Smale condition if any sequence
for which
is bounded and
as
possesses a convergent sequence.
The following deformation theorem is stated in [8].
Theorem 3.2.
Let be a real Banach space and
. Suppose
satisfies Palais-Smale condition. Let
be a given neighborhood of the set
of the critical points of
at a given level
. Then there exists
, as small as we want, and a deformation
such that we denote by
the set
:
(i),
(ii),
(iii).
We state the Mountain Pass Theorem.
Theorem 3.3.
Let be a real Banach space and
satisfy
condition. Suppose that
() there are constants such that
, and
() there is an such that
.
Then possesses a critical value
. Moreover,
can be characterized as

where

4. Critical Point Theory and Multiple Nontrivial Solutions
We investigate the existence of multiple solutions of (1.1) when . We define a functional on
by

Then the functional is well defined in
and the solutions of (1.4) coincide with the critical points of
. Now we investigate the property of functional
.
Lemma 4.1 (cf. [7]).
is continuous and Frechet differentiable at each
with

We will use a variational reduction method to apply the mountain pass theorem.
Let be the two-dimensional subspace of
. Both of them have the same eigenvalue
. Then
for
. Let
be the orthogonal complement of
in
. Let
denote
onto
and
denote
onto
. Then every element
is expressed by

where ,
.
Lemma 4.2.
Let , and
. Let
be given. Then we have that there exists a unique solution
of equation

Let . Then
satisfies a uniform Lipschitz continuous on
with respect to the
norm (also the norm
).
Proof.
Choose and let
Then (4.4) can be written as

Since is a self-adjoint, compact, linear map from
into itself, the eigenvalues of
in
are
, where
or
. Therefore,
is
. Since

the right-hand side of (4.5) defines a Lipschitz mapping because for fixed maps into itself. By the contraction mapping principle, there exists a unique
(also
) for fixed
. Since
is bounded from
to
there exists a unique solution
of (4.4) for given
.
Let

Then . If
and
for any
,
, then

Hence

Since

Therefore, is continuous on
with respect to norm
(also, to
).
Lemma 4.3.
If is defined by
, then
is a continuous Frechet derivative
with respect to
and

If is a critical point of
, then
is a solution of (1.4) and conversely every solution of (1.4) is of this form.
Proof.
Let and set
. If
, then from (4.4)

Since

Let be the two subspaces of
defined as follows:

Given and considering the function
:
defined by

the function has continuous partial Fréchet derivatives
and
with respect to its first and second variables given by

Therefore, let with
and
. Then by Lemma 4.2

If and
, then

Since for any
and
, it is easy to know that

And

It follows that

Therefore, is strictly convex with respect to the second variable.
Similarly, using the fact that for any
, if
and
are in
and
, then

where . Therefore,
is strictly concave with respect to the first variable. From (4.17), it follows that

with equality if and only if .
Since is strictly concave (convex) with respect to its first (second) variable, [9, Theorem
] implies that
is
with respect to
and

Suppose that there exists such that
. From (4.24), it follows that
for all
. Then by Lemma 4.2, it follows that
for any
. Therefore,
is a solution of (1.4).
Conversely, if is a solution of (1.4) and
, then
for any
.
Lemma 4.4.
Let , and
. Then there exists a small open neighborhood
of 0 in
such that
is a strict local minimum of
.
Proof.
For , and
, problem (1.4) has a trivial solution
. Thus we have
. Since the subspace
is orthogonal complement of subspace
, we get
and
. Furthermore,
is the unique solution of (4.4) in
for
. The trivial solution
is of the form
and
, where
is an identity map on
,
is continuous, it follows that there exists a small open neighborhood
of 0 in
such that if
then
,
. By Lemma 4.2,
is the solution of (4.5) for any
. Therefore, if
, then for
we have
. Thus

If , then
. Therefore, in
,

where . It follows that
is a strict local point of minimum of
.
Proposition 4.5.
If , then the equation
admits only the trivial solution
in
.
Proof.
is invariant under
and under the map
. So the spectrum
of
restricted to
contains
in
. The spectrum
of
restricted to
contains
in
. From the symmetry theorem in [10], any solution
of this equation satisfies
. This nontrivial periodic solution is periodic with periodic
. This shows that there is no nontrivial solution of
Lemma 4.6.
Let and
. Then the functional
, defined on
, satisfies the Palais-Smale condition.
Proof.
Let be a Palais-Smale sequence that is
is bounded and
in
. Since
is two-dimensional, it is enough to prove that
is bounded in
.
Let be the solution of (1.4) with
where
. So

By contradiction, we suppose that , also
. Dividing by
and taking
, we get

Since , we get
weakly in
. Since
is a compact operator, passing to a subsequence, we get
strongly in
. Taking the limit of both sides of (4.28), it follows that

with . This contradicts to the fact that for
the following equation

has only the trivial solution by Proposition 4.5. Hence is bounded in
.
We now define the functional on , for
,

The critical points of coincide with solutions of the equation

The above equation () has only the trivial solution and hence
has only one critical point
.
Given , let
be the unique solution of the equation

where . Let us define the reduced functional
on
by
. We note that we can obtain the same results as Lemmas 4.1 and 4.2 when we replace
and
by
and
. We also note that, for
has only the critical point
.
Lemma 4.7.
Let ,
, and
. Then we have
for all
with
.
The proof of the lemma can be found in [1].
Lemma 4.8.
Let ,
, and
. Then we have

for all (certainly for also the norm
).
Proof.
Suppose that it is not true that

Then there exists a sequence in
and a constant
such that

Given , let
be the unique solution of the equation

Let . Then
. By dividing
, we have

By Lemma 4.2, is Lipschitz continuous on
. So the sequence
is bounded in
. Since
(
), it follows that
and
are bounded in
. Since
is a compact operator, there is a subsequence of
converging to some
in
, denoted by itself. Since
is a two-dimensional space, assume that sequence
converges to
with
. Therefore, we can get that the sequence
converges to an element
in
.
On the other hand, since , dividing this inequality by
, we get

By Lemma 4.2, it follows that for any

If we set in (4.40) and divide by
, then we obtain

Let be arbitrary. Dividing (4.40) by
and letting
, we obtain

where Then (4.42) can be written in the form
for all
. Put
. Letting
in (4.41), we obtain

where we have used (4.42). Hence

Letting in (4.39), we obtain

Since , this contradicts to the fact that
for all
. This proves that
.
Now we state the main result in this paper.
Theorem 4.9.
Let ,
, and
. Then there exist at least three solutions of the equation

two of which are nontrivial solutions.
Proof.
We remark that is the trivial solution of problem (1.4). Then
is a critical point of functional
. Next we want to find others critical points of
which are corresponding to the solutions of problem (1.4).
By Lemma 4.4, there exists a small open neighborhood of 0 in
such that
is a strict local point of minimum of
. Since
from Lemma 4.8 and
is a two-dimensional space, there exists a critical point
of
such that

Let be an open neighborhood of
in
such that
. Since
, we can choose
such that
. Since
satisfies the Palais-Smale condition, by the Mountain Pass Theorem (Theorem 3.3), there is a critical value

where
If , then there exists a critical point
of
at level
such that
, 0 ( since
and
). Therefore, in case
, the functional
has also at least 3 critical points
.
If , then define

where . Hence,

That is . By contradiction, assume
. Use the functional
for the deformation theorem (Theorem 4.9) and taking
. We choose
such that
. From the deformation theorem (Theorem 3.2),
and

which is a contradiction. Therefore, there exists a critical point of
at level
such that
, 0, which means that (1.4) has at least three critical points. Since
, these two critical points coincide with two nontrivial period solutions of problem (1.4).
5. Nontrivial Solutions for the Beam System
In this section, we investigate the existence of multiple nontrivial solutions for perturbations
of the beam system with Dirichlet boundary condition

where and the nonlinearity
crosses the eigenvalues of the beam operator.
Theorem 5.1.
Let ,
, and
. Then beam system (5.1) has at least three solutions
, two of which are nontrivial solutions.
Proof.
From problem (5.1), we get the equation

where the nonlinearity
Let . Then the above equation is equivalent to

Since ,
, and
, the above equation has at least three solutions, two of which are nontrivial solutions, say
. Hence we get the solutions
of problem (5.1) from the following systems:

where and
. When
, from the above equation, we get the trivial solution
. When
, from the above equation, we get the nontrivial solutions
,
.Therefore, system(5.1) has at least three solutions
, two of which are nontrivial solutions.
Theorem 5.2.
Let , and
. Then system (5.1) has only the trivial solution
.
Proof.
From problem (5.1), we get the equation

where the nonlinearity
Let . Then the above equation is equivalent to

Since , and
, by Theorem 2.2, the above equation has the trivial solution. Hence we have the trivial solution
of problem (5.1) from the following system:

From (5.9), we get the trivial solution .
References
Choi Q, Jung T, McKenna PJ: The study of a nonlinear suspension bridge equation by a variational reduction method. Applicable Analysis 1993,50(1-2):73-92. 10.1080/00036819308840185
Lazer AC, McKenna PJ: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Review 1990,32(4):537-578. 10.1137/1032120
Tarantello G: A note on a semilinear elliptic problem. Differential and Integral Equations 1992,5(3):561-565.
Micheletti AM, Pistoia A: Multiplicity results for a fourth-order semilinear elliptic problem. Nonlinear Analysis: Theory, Methods & Applications 1998,31(7):895-908. 10.1016/S0362-546X(97)00446-X
Micheletti AM, Pistoia A: Nontrivial solutions for some fourth order semilinear elliptic problems. Nonlinear Analysis: Theory, Methods & Applications 1998,34(4):509-523. 10.1016/S0362-546X(97)00596-8
Lazer AC, McKenna PJ: Global bifurcation and a theorem of Tarantello. Journal of Mathematical Analysis and Applications 1994,181(3):648-655. 10.1006/jmaa.1994.1049
Choi Q, Jung T: An application of a variational reduction method to a nonlinear wave equation. Journal of Differential Equations 1995,117(2):390-410. 10.1006/jdeq.1995.1058
Rabinobitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Mathematical Science Regional Conference Series, no. 65. American Mathematical Society, Providence, RI, USA; 1984.
Amann H: Saddle points and multiple solutions of differential equations. Mathematische Zeitschrift 1979,169(2):127-166. 10.1007/BF01215273
Lazer AC, McKenna PJ: A symmetry theorem and applications to nonlinear partial differential equations. Journal of Differential Equations 1988,72(1):95-106. 10.1016/0022-0396(88)90150-7
Acknowledgments
This work (Choi) was supported by Inha University Research Grant. The authors appreciate very much the referee's corrections and revisions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Jung, T., Choi, QH. Nontrivial Solutions of the Asymmetric Beam System with Jumping Nonlinear Terms. Bound Value Probl 2010, 728101 (2010). https://doi.org/10.1155/2010/728101
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/728101