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Infinitely Many Periodic Solutions for Variable Exponent Systems
Journal of Inequalities and Applications volume 2009, Article number: 714179 (2009)
Abstract
We mainly consider the system in
,
in
, where
are periodic functions, and
is called
-Laplacian. We give the existence of infinitely many periodic solutions under some conditions.
1. Introduction
The study of differential equations and variational problems with variable exponent growth conditions has been a new and interesting topic. Many results have been obtained on this kind of problems, for example [1–18]. On the applied background, we refer to [1, 3, 11, 18]. In this paper, we mainly consider the existence of infinitely many periodic solutions for the system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ1_HTML.gif)
where are functions. The operator
is called one-dimensional
-Laplacian. Especially, if
(a constant) and
(a constant), then
is the well-known constant exponent system.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_IEq14_HTML.gif)
is called a solution of , if
,
and
are absolute continuous and satisfy
almost every where.
In [19], the authors consider the existence of positive weak solutions for the following constant exponent problems:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ2_HTML.gif)
The first eigenfunction is used to construct the subsolution of constant exponent problems successfully. Under the condition that is large enough and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ3_HTML.gif)
the authors give the existence of positive solutions for problem .
In [20], the author considers the existence and nonexistence of positive weak solution to the following constant exponent elliptic system:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ4_HTML.gif)
The first eigenfunction is used to construct the subsolution of constant exponent problems successfully.
Because of the nonhomogeneity of -Laplacian,
-Laplacian problems are more complicated than those of
-Laplacian. Maybe the first eigenvalue and the first eigenfunction of
-Laplacian do not exist (see [6]). Even if the first eigenfunction of
-Laplacian exists, because of the nonhomogeneity of
-Laplacian, the first eigenfunction cannot be used to construct the subsolution of
-Laplacian problems.
There are many papers on the existence of periodic solutions for -Laplacian elliptic systems, for example [21–24]. The results on the periodic solutions for variable exponent systems are rare. Through a new method of constructing sub-supersolution, this paper gives the existence of infinitely many periodic solutions for problem
.
2. Main Results and Proofs
At first, we give an existence of positive solutions for variable exponent systems on bounded domain via sub-super-solution method. The result itself has dependent value.
Denote . Let us consider the existence of positive solutions of the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ5_HTML.gif)
Write ,
, for any
. Assume that
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ6_HTML.gif)
,
are
, monotone functions such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ7_HTML.gif)
For any positive constant , there are
.
.
, and
are odd functions such that
,
and
are even, and
is a periodic of
and
, namely,
Note. In [14], the present author discussed the existence of solutions of , under the conditions that
is radial,
, and
. Because of the nonhomogeneity of variable exponent problems, variable exponent problems are more complicated than constant exponent problems, and many results and methods for constant exponent problems are invalid for variable exponent problems. In many cases, the radial symmetric conditions are effective to deal with variable exponent problems. There are many results about the radial variable exponent problems (see [4, 14, 16]), but the following Theorem 2.1 does not assume any symmetric conditions.
We will establish.
Theorem 2.1.
If hold, then
possesses a positive solution, when
is sufficiently large.
Proof.
If we can construct a positive subsolution and supersolution
of
, namely,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ8_HTML.gif)
which satisfy and
, then
possesses a positive solution (see [5]).
Step 1.
We will construct a subsolution of .
Denfine
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ9_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_IEq71_HTML.gif)
and satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_IEq73_HTML.gif)
and satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ12_HTML.gif)
then , and
. It is easy to see that
and
,
. Obviously,
is continuous about
.
In the following, we will prove that is a subsolution for
. By computation,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ13_HTML.gif)
If is sufficiently large, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ14_HTML.gif)
As is a constant and only depends on
and
, when
is large enough, we have
. Since
and
is monotone, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ15_HTML.gif)
According to (), when
are large enough, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ16_HTML.gif)
where are dependent on
,
,
,
and
, and they are independent on
. Since
, when
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ17_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ18_HTML.gif)
Obviously
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ19_HTML.gif)
When is large enough, from (2.7) we can see that
is large enough. Similar to the discussion of the above, we can conclude
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ21_HTML.gif)
Since combining (2.11), (2.14), (2.15), (2.16) and (2.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ22_HTML.gif)
Similarly, when is large enough, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ23_HTML.gif)
Then is a subsolution of
.
Step 2.
We will construct a supersolution of .
Let be a solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ24_HTML.gif)
where is a positive constant and
.
Obviously, there exists such that
. Note that
is dependent on
. Denote
. It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ25_HTML.gif)
Let us consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ26_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ27_HTML.gif)
We will prove that is a supersolution for
. From
and (2.23), when
is large enough, we can easily see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ28_HTML.gif)
Since and
, when
is large enough, according to (2.21) and (2.23), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ29_HTML.gif)
This means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ30_HTML.gif)
According to (2.24) and (2.26), we can conclude that is a supersolution for
, when
is large enough.
Step 3.
We will prove that and
.
Obviously, when is large enough, we can easily see that
is large enough, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ31_HTML.gif)
Let us consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ32_HTML.gif)
It is easy to see that is a subsolution of (2.28), when
is large enough. Obviously, we can see that
is a supersolution of (2.28), and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ33_HTML.gif)
According to the comparison principle (see [12]), we can see that .
Let us consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ34_HTML.gif)
It is easy to see that is a subsolution of (2.30), when
is large enough. Obviously, we can see that
is a supersolution of (2.30), and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ35_HTML.gif)
According to the comparison principle (see [12]), we can see that .
Thus, we can conclude that and
, when
is sufficiently large. This completes the proof.
Theorem 2.2.
If hold, then
has infinitely many periodic solutions.
Proof.
Let . According to Theorem 2.1, we can conclude that there exists an integer
which is large enough such that
has a positive solution
for any integer
. Since
and
are even, and
are odd, then
is a negative solution of
. We can define a
function
on
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F714179/MediaObjects/13660_2008_Article_1994_Equ36_HTML.gif)
We extend as
where
is an integer such that
. It is easy to see that
is a solution of
and the periodic of
is
. This completes the proof.
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Acknowledgment
Partly supported by the National Science Foundation of China (10701066 & 10671084) and China Postdoctoral Science Foundation (20070421107) and the Natural Science Foundation of Henan Education Committee (2008-755-65 & 2009A120005) and the Natural Science Foundation of Jiangsu Education Committee (08KJD110007).
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Guo, X., Lu, M. & Zhang, Q. Infinitely Many Periodic Solutions for Variable Exponent Systems. J Inequal Appl 2009, 714179 (2009). https://doi.org/10.1155/2009/714179
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DOI: https://doi.org/10.1155/2009/714179