- Research Article
- Open access
- Published:
Multiple Solutions for a Class of
-Laplacian Systems
Journal of Inequalities and Applications volume 2009, Article number: 191649 (2009)
Abstract
We study the multiplicity of solutions for a class of Hamiltonian systems with the -Laplacian. Under suitable assumptions, we obtain a sequence of solutions associated with a sequence of positive energies going toward infinity.
1. Introduction and Main Results
Since the space and
were thoroughly studied by Kováčik and Rákosník [1], variable exponent Sobolev spaces have been used in the last decades to model various phenomena. In [2], Růžička presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids.
In recent years, the differential equations and variational problems with -growth conditions have been studied extensively; see for example [3–6]. In [7], De Figueiredo and Ding discussed the multiple solutions for a kind of elliptic systems on a smooth bounded domain. Motivated by their work, we will consider the following sort of
-Laplacian systems with "concave and convex nonlinearity":
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ1_HTML.gif)
where is a bounded domain,
is continuous on
and satisfies
, and
is a
function. In this paper, we are mainly interested in the class of Hamiltonians
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ2_HTML.gif)
where Here we denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ3_HTML.gif)
and denote by the fact that
Throughout this paper,
satisfies the following conditions:
(H1) Writing
(H2) there exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ4_HTML.gif)
where is positive constant;
(H3) there exist with
, and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ5_HTML.gif)
when
As [8,Lemma 1.1], from assumption (H3), there exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ6_HTML.gif)
for any We can also get that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ7_HTML.gif)
for any In this paper, we will prove the following result.
Theorem 1.1.
Assume that hypotheses (H1)–(H3) are fulfilled. If is even in
, then problem (1.1) has a sequence of solutions
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ8_HTML.gif)
as
2. Preliminaries
First we recall some basic properties of variable exponent spaces and variable exponent Sobolev spaces
where
is a domain. For a deeper treatment on these spaces, we refer to [1, 9–11].
Let be the set of all Lebesgue measurable functions
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ9_HTML.gif)
The variable exponent space is the class of all functions
such that
Under the assumption that
is a Banach space equipped with the norm (2.1).
The variable exponent Sobolev space is the class of all functions
such that
and it can be equipped with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ10_HTML.gif)
For if we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ11_HTML.gif)
then and
are equivalent norms on
By we denote the subspace of
which is the closure of
with respect to the norm (2.2) and denote the dual space of
by
We know that if
is a bounded domain,
and
are equivalent norms on
Under the condition is a separable and reflexive Banach space, then there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ12_HTML.gif)
In the following, we will denote that where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ13_HTML.gif)
For any define the norm
For any
set
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ14_HTML.gif)
denote the complement of in
by
3. The Proof of Theorem 1.1
Definition 3.1.
We say that is a weak solution of problem (1.1), that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ15_HTML.gif)
In this section, we denote that for any
, and
is positive constant, for any
Lemma 3.2.
Any sequence
, that is,
and
as
is bounded.
Proof.
Let be sufficiently small such that
Let be such that
as
We get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ16_HTML.gif)
As by the Young inequality, we can get that for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ17_HTML.gif)
Let be sufficiently small such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ18_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ19_HTML.gif)
Note that by the Young inequality, for any
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ20_HTML.gif)
Let be sufficiently small such that
and
then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ21_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ22_HTML.gif)
and for being large enough, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ23_HTML.gif)
It is easy to know that if and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ24_HTML.gif)
thus we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ25_HTML.gif)
then are bounded. Similarly, if
or
we can also get that
are bounded. It is immediate to get that
is bounded in
Lemma 3.3.
Any sequence contains a convergent subsequence.
Proof.
Let be a
sequence. By Lemma 3.2, we obtain that
is bounded in
As
is reflexive, passing to a subsequence, still denoted by
we may assume that there exists
such that
weakly in
Then we can get
weakly in
. Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ26_HTML.gif)
It is easy to get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ27_HTML.gif)
and in
in
as
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ28_HTML.gif)
as By condition (H2), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ29_HTML.gif)
It is immediate to get that are bounded and
then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ30_HTML.gif)
as Similar to [3, 4], we divide
into two parts:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ31_HTML.gif)
On we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ32_HTML.gif)
then On
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ33_HTML.gif)
Thus we get Then
in
as
Similarly,
in
Lemma 3.4.
There exists such that
for all
with
Proof.
For any we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ34_HTML.gif)
In the following, we will consider
-
(i)
If
We have
(321)
-
(ii)
If
Note that
For any
there exists
which is an open subset of
such that
(322)
then is an open covering of
As
is compact, we can pick a finite subcovering
for
Thus there exists a sequence of open set
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ37_HTML.gif)
for Denote that
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ38_HTML.gif)
where As
is a finite dimensional space, we have
for
We denote by the maximum of polynomial
on
for
Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ39_HTML.gif)
for and
where
Let If
we get
or
(i)If It is easy to verify that there exists at least
such that
thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ40_HTML.gif)
(ii)If We obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ41_HTML.gif)
Now we get the result.
Lemma 3.5.
There exist and
such that
for any
with
Proof.
For By condition (H2), there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ42_HTML.gif)
Let we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ43_HTML.gif)
Denote that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ44_HTML.gif)
thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ45_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ46_HTML.gif)
By [5, Lemma 3.3], we get that as
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ47_HTML.gif)
when is sufficiently large and
It is easy to get that
as
Lemma 3.6.
I is bounded from above on any bounded set of
Proof.
For We get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ48_HTML.gif)
By conditions (H2) and (H3), we know that if and if
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191649/MediaObjects/13660_2008_Article_1912_Equ49_HTML.gif)
and it is easy to get the result.
Proof.
By Lemmas 3.2–3.6 above, and [7, Proposition 2.1 and Remark 2.1], we know that the functional has a sequence of critical values
as
Now we complete the proof.
References
Kováčik O, Rákosník J: On spaces and . Czechoslovak Mathematical Journal 1991,41(116)(4):592–618.
Růžička M: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics. Volume 1748. Springer, Berlin, Germany; 2000:xvi+176.
Chabrowski J, Fu Y: Existence of solutions for -Laplacian problems on a bounded domain. Journal of Mathematical Analysis and Applications 2005,306(2):604–618. 10.1016/j.jmaa.2004.10.028
Chabrowski J, Fu Y: Corrigendum to: "Existence of solutions for -Laplacian problems on a bounded domain". Journal of Mathematical Analysis and Applications 2006,323(2):1483. 10.1016/j.jmaa.2005.11.036
Fan X, Han X: Existence and multiplicity of solutions for -Laplacian equations in . Nonlinear Analysis: Theory, Methods & Applications 2004,59(1–2):173–188.
Mihăilescu M, Rădulescu V: A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proceedings of the Royal Society of London. Series A 2006,462(2073):2625–2641. 10.1098/rspa.2005.1633
De Figueiredo DG, Ding YH: Strongly indefinite functionals and multiple solutions of elliptic systems. Transactions of the American Mathematical Society 2003,355(7):2973–2989. 10.1090/S0002-9947-03-03257-4
Felmer PL: Periodic solutions of "superquadratic" Hamiltonian systems. Journal of Differential Equations 1993,102(1):188–207. 10.1006/jdeq.1993.1027
Edmunds DE, Lang J, Nekvinda A: On norms. Proceedings of the Royal Society of London. Series A 1999,455(1981):219–225. 10.1098/rspa.1999.0309
Edmunds DE, Rákosník J: Sobolev embeddings with variable exponent. Studia Mathematica 2000,143(3):267–293.
Fan X, Zhao Y, Zhao D: Compact imbedding theorems with symmetry of Strauss-Lions type for the space . Journal of Mathematical Analysis and Applications 2001,255(1):333–348. 10.1006/jmaa.2000.7266
Acknowledgments
This work is supported by Science Research Foundation in Harbin Institute of Technology (HITC200702) and The Natural Science Foundation of Heilongjiang Province (A2007-04).
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
Fu, Y., Zhang, X. Multiple Solutions for a Class of -Laplacian Systems.
J Inequal Appl 2009, 191649 (2009). https://doi.org/10.1155/2009/191649
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/191649