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Periodic Problem with a Potential Landesman Lazer Condition
Boundary Value Problems volume 2010, Article number: 586971 (2010)
Abstract
We prove the existence of a solution to the periodic nonlinear second-order ordinary differential equation with damping ,
,
. We suppose that
, the nonlinearity
satisfies the potential Landesman Lazer condition and prove that a critical point of a corresponding energy functional is a solution to this problem.
1. Introduction
Let us consider the nonlinear problem

where , the nonlinearity
is a Caratheodory function and
.
To state an existence result to (1.1) Amster [1] assumes that is a nondecreasing function (see also [2]). He supposes that the nonlinearity
satisfies the growth condition
,
for
,
,
where
is the first eigenvalue of the problem
,
and there exist
such that
An interval
is centered in
with the radius
where
,
and
is a solution to the problem
with
.
In [3, 4] authors studied (1.1) with a constant friction term and results with repulsive singularities were obtained in [5, 6].
In this paper we present new assumptions, we suppose that the friction term has zero mean value

the nonlinearity is bounded by a
function and satisfies the following potential Landesman-Lazer condition (see also [7, 8])

where,
,
and
To obtain our result we use variational approach even if the linearization of the periodic problem (1.1) is a non-self-adjoint operator.
2. Preliminaries
Notation 2.
We will use the classical space of functions whose
th derivative is continuous and the space
of measurable real-valued functions whose
th power of the absolute value is Lebesgue integrable. We denote
the Sobolev space of absolutely continuous functions
such that
and
with the norm
. By a solution to (1.1) we mean a function
such that
is absolutely continuous,
satisfies the boundary conditions and (1.1) is satisfied a.e. in
.
We denote and we study (1.1) by using variational methods. We investigate the functional
, which is defined by

where

We say that is a critical point of
, if

We see that every critical point of the functional
satisfies

for all .
Now we prove that any critical point of the functional is a solution to (1.1) mentioned above.
Lemma 2.1.
Let the condition (1.2) be satisfied. Then any critical point of the functional is a solution to (1.1).
Proof.
Setting in (2.4) we obtain

We denote

then previous equality (2.5) implies and by parts in (2.4) we have

for all Hence there exists a constant
such that

on . The condition (1.2) implies
and from (2.8) we get
Using
and differentiating equality (2.8) with respect to
we obtain

Thus is a solution to (1.1).
We say that satisfies the Palais-Smale condition (PS) if every sequence
for which
is bounded in
and
(as
possesses a convergent subsequence.
To prove the existence of a critical point of the functional we use the Saddle Point Theorem which is proved in Rabinowitz [9] (see also [10]).
Theorem 2.2 (Saddle Point Theorem).
Let ,
and
. Let
be a functional such that
and
(a) there exists a bounded neighborhood of
in
and a constant
such that
,
(b) there is a constant such that
,
(c) satisfies the Palais-Smale condition (PS).
Then, the functional has a critical point in
.
3. Main Result
We define

Assume that the following potential Landesman-Lazer type condition holds:

We also suppose that there exists a function such that

Theorem 3.1.
Under the assumptions (1.2), (3.2), (3.3), problem (1.1) has at least one solution.
Proof.
We verify that the functional satisfies assumptions of the Saddle Point Theorem 2.2 on
, then
has a critical point
and due to Lemma 2.1ââ
is the solution to (1.1).
It is easy to see that . Let
then
and
.
In order to check assumption (a), we prove

by contradiction. Then, assume on the contrary there is a sequence of numbers such that
and a constant
satisfying

From the definition of and from (3.5) it follows

We note that from (3.2) it follows there exist constants ,
and functions
such that
,
for a.e.
and for all
,
, respectively. We suppose that for this moment
. Using (3.6) and Fatou's lemma we obtain

a contradiction to (3.2). We proceed for the case Then assumption (a) of Theorem 2.2 is verified.
(b) Now we prove that is bounded from below on
. For
, we have

and assumption (3.3) implies

Hence and due to compact imbedding we obtain

Since the function is strictly positive equality (3.10) implies that the functional
is bounded from below.
Using (3.4), (3.10) we see that there exists a bounded neighborhood of
in
, a constant
such that
, and there is a constant
such that
.
In order to check assumption (c), we show that satisfies the Palais-Smale condition. First, we suppose that the sequence
is unbounded and there exists a constant
such that


Let be an arbitrary sequence bounded in
. It follows from (3.12) and the Schwarz inequality that

From (3.3) we obtain

Put and
then (3.13), (3.14) imply

Due to compact imbedding and (3.15) we have
in
,
. Suppose that
and set
in (3.13), we get

Because the nonlinearity is bounded (assumption (3.3)) and
the second integral in previous equality (3.16) converges to zero. Therefore

Now we divide (3.11) by . We get

Equalities (3.17), (3.18) imply

Because ,
. Using Fatou's lemma and (3.19) we conclude

a contradiction to (3.2). We proceed for the case similarly. This implies that the sequence
is bounded. Then there exists
such that
in
,
in
,
(taking a subsequence if it is necessary). It follows from equality (3.13) that

The strong convergence in
and the assumption (3.3) imply

If we set ,
in (3.21) and subtract these equalities, then using (3.22) we have

Hence we obtain the strong convergence in
. This shows that
satisfies the Palais-Smale condition and the proof of Theorem 3.1 is complete.
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Acknowledgment
This work was supported by Research Plan MSM 4977751301.
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Tomiczek, P. Periodic Problem with a Potential Landesman Lazer Condition. Bound Value Probl 2010, 586971 (2010). https://doi.org/10.1155/2010/586971
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DOI: https://doi.org/10.1155/2010/586971