Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 714179, 10 pages
doi:10.1155/2009/714179
Research Article
Infinitely Many Periodic Solutions for
Variable Exponent Systems
Xiaoli Guo,1 Mingxin Lu,2 and Qihu Zhang1
1
Department of Mathematics and Information Science, Zhengzhou University of Light Industry,
Zhengzhou, Henan 450002, China
2
Department of Information Management, Nanjing University, Nanjing 210093, China
Correspondence should be addressed to Qihu Zhang, zhangqh1999@yahoo.com.cn
Received 4 December 2008; Accepted 14 April 2009
Recommended by Marta Garcia-Huidobro
We mainly consider the system −Δpx u fv hu in R, −Δqx v gu ωv in R, where
1 < px, qx ∈ C1 R are periodic functions, and −Δpx u −|u |px−2 u is called px-Laplacian.
We give the existence of infinitely many periodic solutions under some conditions.
Copyright q 2009 Xiaoli Guo et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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
P ⎧
⎨−Δpx u fv hu
⎩−Δ
qx v
in R,
1.1
gu ωv in R,
where px, qx ∈ C1 R are functions. The operator −Δpx u −|u |px−2 u is called onedimensional px-Laplacian. Especially, if px ≡ p a constant and qx ≡ q a constant,
then P is the well-known constant exponent system.
u, v is called a solution of P , if u, v ∈ C1 R, |u |px−2 u and |v |px−2 v are absolute
continuous and satisfy P almost every where.
2
Journal of Inequalities and Applications
In 19, the authors consider the existence of positive weak solutions for the following
constant exponent problems:
⎧
⎪
−Δ u λfv,
⎪
⎪ p
⎨
I −Δp v λgu,
⎪
⎪
⎪
⎩
u v 0,
in Ω,
in Ω,
1.2
on ∂Ω.
The first eigenfunction is used to construct the subsolution of constant exponent problems
successfully. Under the condition that λ is large enough and
1/ p−1 f M gu lim
u → ∞
up−1
0,
for every M > 0,
1.3
the authors give the existence of positive solutions for problem I.
In 20, the author considers the existence and nonexistence of positive weak solution
to the following constant exponent elliptic system:
⎧
⎪
−Δ u λuα vγ ,
⎪
⎪ p
⎨
II −Δq v λuδ vβ ,
⎪
⎪
⎪
⎩
u v 0,
in Ω,
in Ω,
1.4
on ∂Ω.
The first eigenfunction is used to construct the subsolution of constant exponent problems
successfully.
Because of the nonhomogeneity of px-Laplacian, px-Laplacian problems are more
complicated than those of p-Laplacian. Maybe the first eigenvalue and the first eigenfunction
of px-Laplacian do not exist see 6. Even if the first eigenfunction of px-Laplacian
exists, because of the nonhomogeneity of px-Laplacian, the first eigenfunction cannot be
used to construct the subsolution of px-Laplacian problems.
There are many papers on the existence of periodic solutions for p-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 P .
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.
Journal of Inequalities and Applications
3
Denote ΩR −R, R. Let us consider the existence of positive solutions of the
following:
⎧
⎪
−Δpx u fv hu, in ΩR ,
⎪
⎪
⎨
P1 −Δqx v gu ωv, in ΩR ,
⎪
⎪
⎪
⎩
u v 0,
on ∂ΩR .
2.1
Write z supx∈R zx, z− infx∈R zx, for any z ∈ CR. Assume that
H1 px, qx ∈ C1 R satisfy
sup
p x
< ∞,
1 < p− ≤ p < ∞,
1 < q− ≤ q < ∞,
sup
q x
< ∞.
2.2
H2 f, g, h, ω : 0, ∞ → R are C1 , monotone functions such that
lim ft lim gt lim ht lim ωt ∞.
t → ∞
t → ∞
t → ∞
H3 For any positive constant M, there are limt → ∞ fMgt1/q
−
p −1
H4 limt → ∞ ht/t
−
q −1
limt → ∞ ωt/t
2.3
t → ∞
−
−1
/tp
−
−1
0.
0.
H5 f, g, h, and ω are odd functions such that f0 g0 h0 ω0 0, px
and qx are even, and T is a periodic of p and q, namely, px px T , qx qx T , for all x ∈ R.
Note. In 14, the present author discussed the existence of solutions of P1 , under the
conditions that P1 is radial, px qx, and h ω ≡ 0. 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 H1 –H4 hold, then P1 possesses a positive solution, when R is sufficiently large.
Proof. If we can construct a positive subsolution φ1 , φ2 and supersolution z1 , z2 of P1 ,
namely,
−Δpx φ1 ≤ f φ2 h φ1 ,
−Δqx φ2 ≤ g φ1 ω φ2 ,
−Δpx z1 ≥ fz2 hz1 ,
−Δqx z2 ≥ gz1 ωz2 ,
for a.e. x ∈ R,
for a.e. x ∈ R,
which satisfy φ1 ≤ z1 and φ2 ≤ z2 , then P1 possesses a positive solution see 5.
2.4
4
Journal of Inequalities and Applications
Step 1. We will construct a subsolution of P1 .
Denfine
⎧ −k x−R
− 1, R − a < x ≤ R,
e 2
⎪
⎪
⎪
⎪
⎪
⎪
R−a
⎪
⎪
ak pR−a−1/pr−1 π 1/pr−1
⎪
⎪
ak2
2
⎪
sin
ε
−
1
e
e
−
−
a
dr,
k
r
R
2
2
⎪
⎪
2
⎪
x
⎪
⎪
⎪
π
⎪
⎪
⎪
R−a−
< x ≤ R − a,
⎪
⎪
2ε
2
⎪
⎪
⎪
⎪
R−a
⎪
⎪
⎪ ak
ak pR−a−1/pr−1 π 1/pr−1
⎪
⎨e 2 −1
sin ε2 r −R−a
dr,
k2 e 2
2
φ1 x R−a−π/2ε2
⎪
⎪
π
π
⎪
⎪
−R a <x ≤R−a−
,
⎪
⎪
2ε1
2ε2
⎪
⎪
⎪
⎪
⎪
x
⎪
1/pr−1
⎪
ak p−Ra−1/pr−1 π
⎪
ak1
⎪
1
⎪
e
sin
−
ε
−
1
e
−
a
dr,
k
r
−R
1
1
⎪
⎪
2
⎪
−Ra
⎪
⎪
⎪
π
⎪
⎪
−R a ≤ x ≤ −R a ,
⎪
⎪
⎪
2ε
1
⎪
⎪
⎪
⎩ k1 xR
− 1, −R ≤ x < −R a.
e
⎧ −k x−R
− 1, R − a < x ≤ R,
e 4
⎪
⎪
⎪
⎪
⎪
⎪
R−a
⎪
⎪
ak qR−a−1/qr−1 π 1/qr−1
⎪
⎪
ak4
4
⎪
sin
ε
−
1
e
e
−
−
a
dr,
k
r
R
4
4
⎪
⎪
2
⎪
x
⎪
⎪
⎪
π
⎪
⎪
⎪
< x ≤ R − a,
R−a−
⎪
⎪
2ε4
⎪
⎪
⎪
⎪
R−a
⎪
⎪
⎪
ak qR−a−1/qr−1 π 1/qr−1
⎪
⎨eak4 −1
sin ε4 r −R−a
dr,
k4 e 4
2
φ2 x R−a−π/2ε4
⎪
⎪
π
π
⎪
⎪
−R a <x ≤R−a−
,
⎪
⎪
2ε
2ε
⎪
3
4
⎪
⎪
⎪
⎪
x
⎪
1/qr−1
⎪
ak q−Ra−1/qr−1 π
⎪
ak
⎪
3
⎪e − 1 k3 e 3
sin
− ε3 r − −R a
dr,
⎪
⎪
2
⎪
−Ra
⎪
⎪
⎪
π
⎪
⎪
−R a ≤ x ≤ −R a ,
⎪
⎪
⎪
2ε
3
⎪
⎪
⎪
⎩ k3 xR
− 1, −R ≤ x < −R a,
e
2.5
where
inf qx − 1
inf px − 1
a min , ,
4 sup
p x
1 4 sup
q x
1
−p −aki p
εi ki
e
,
i 1, 2,
−q −aki q
εi ki
e
,
b min f0, g0, h0, ω0, −1 ,
i 3, 4;
R>
π
,
εi
i 1, 2, 3, 4,
2.6
Journal of Inequalities and Applications
5
k1 and k2 satisfy
eak2 − 1 R−a
k2 eak2
R−a−π/2ε2
eak1 −1
pR−a−1/pr−1 π 1/pr−1
sin ε2 r − R − a dr
2
2.7
−Raπ/2ε1
1/pr−1
p−Ra−1/pr−1 π
k1 eak1
−ε1 r −−Ra
sin
dr,
2
−Ra
k3 and k4 satisfy
eak4 − 1 R−a
k4 eak4
qR−a−1/qr−1 R−a−π/2ε4
eak3 − 1 −Raπ/2ε3 −Ra
k3 eak3
π 1/qr−1
sin ε4 r − R − a dr
2
q−Ra−1/qr−1 2.8
1/qr−1
π
sin
− ε3 r − −R a
dr,
2
then φ1 x ∈ C−R, R, and φ2 x ∈ C−R, R. It is easy to see that φi ≥ 0 and φi ∈
−p
C1 −R, R, i 1, 2. Obviously, εi ki e−aki p is continuous about ki .
In the following, we will prove that φ1 , φ2 is a subsolution for P1 . By computation,
− Δpx φ1 ⎧
px−1 ⎪
−k2 px−1 p x ln k 2 −k2 p xx−R ,
k2 e−k2 x−R
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ε k eak2 pR−a−1 cos ε x − R − a π ,
⎪
2
2
2
⎪
⎪
2
⎪
⎪
⎪
⎨
0,
⎪
⎪
⎪
⎪
⎪ p−Ra−1 π
⎪
⎪
⎪
− ε1 x − −R a ,
cos
⎪ε1 k1 eak1
⎪
⎪
2
⎪
⎪
⎪
⎪
px−1
⎩
k1 ek1 xR
−k1 px−1 −p x ln k1 −k1 p xxR ,
R − a < x ≤ R,
R−
π
< x < R − a,
2ε2
−Ra
π
π
< x < R−a−
,
2ε1
2ε2
−R a < x < −R a,
−R ≤ x < −R a.
2.9
If k2 is sufficiently large, we have
ln k2
−Δpx φ1 ≤ −k2 inf px − 1 − sup
p x
R−r
≤ −k2 a,
k2
∀x ∈ R − a, R.
2.10
As a is a constant and only depends on px and qx, when k2 is large enough, we
have −k2 a < 2b. Since φ1 x ≥ 0 and f h is monotone, we have
−Δpx φ1 ≤ 2b ≤ f0 h0 ≤ f φ2 h φ1 ,
R − a < x ≤ R.
2.11
6
Journal of Inequalities and Applications
According to H2 , when ki are large enough, we have
f eaki − 1 ≥ 1,
g eaki − 1 ≥ 1,
h eaki − 1 ≥ 1,
ω eaki − 1 ≥ 1,
i 1, 2, 3, 4,
2.12
where ki are dependent on f, g, h, ω, p and q, and they are independent on R. Since ε2 −p
k2 e−ak2 p , when x ∈ R − a − π/2ε, R − a, we have
pR−1−1
π
p
≤ ε2 k2 eak2 p 1.
cos εx − R − a −Δpx φ1 ε2 k2 eak2
2
2.13
Then we have
−Δpx φ1 ≤ 1 ≤ f φ2 h φ1 ,
π
< x < R − a.
2ε2
2.14
π
π
<x <R−a−
.
2ε1
2ε2
2.15
R−a−
Obviously
−Δpx φ1 0 ≤ 1 ≤ f φ2 h φ1 ,
−Ra
When k2 is large enough, from 2.7 we can see that k1 is large enough. Similar to the
discussion of the above, we can conclude
−Δpx φ1 ≤ 1 ≤ f φ2 h φ1 ,
−Δpx φ1 ≤ f0 h0 ≤ f φ2 h φ1 ,
− R a < x < −R a π
,
2ε1
− R < x < −R a.
2.16
2.17
Since φi x ∈ C1 −R, R, combining 2.11, 2.14, 2.15, 2.16 and 2.17, we have
−Δpx φ1 ≤ f φ2 h φ1 ,
for a.e. x ∈ −R, R.
2.18
for a.e. x ∈ −R, R.
2.19
Similarly, when k4 is large enough, we have
−Δqx φ2 ≤ g φ1 ω φ2 ,
Then φ1 , φ2 is a subsolution of P1 .
Step 2. We will construct a supersolution of P1 .
Let z1 be a solution of
px−2 z1 2μ, z1 R 0 z1 −R,
− z1 where μ is a positive constant and μ > 1.
2.20
Journal of Inequalities and Applications
7
R
Obviously, there exists x0 ∈ ΩR such that z1 x x |r − x0 2μ|1/pr−1−1 2μr − x0 dr.
Note that x0 is dependent on μ. Denote β β2μ max|x|≤R z1 x. It is easy to see that
1 1/p −1
−
μ
≤ β 2μ ≤ Cμ1/p −1 ,
C
where C ≥ 1 is a positive constant.
2.21
Let us consider
−px z1 2μ in ΩR ,
in ΩR ,
−qx z2 2g β 2μ
2.22
z1 z2 0 on ∂ΩR .
Similarly, we have
1/q− −1
1 1/q −1
≤ max z2 x ≤ C g β 2μ
.
g β 2μ
C
|x|≤R
We will prove that z1 , z2 is a supersolution for P1 . From limt → ∞ ωt/tq
2.23, when μ is large enough, we can easily see that
−Δqx z2 2g β 2μ ≥ gz1 ωz2 .
−
Since limt → ∞ fCg2t1/q −1 /tp −1 0 and limt → ∞ ht/tp
enough, according to 2.21 and 2.23, we have
−
2.23
−
−1
0 and
2.24
−
−1
0, when μ is large
−
1 p −1
1/q− −1
h β 2μ .
β 2μ
≥ f C g β 2μ
2μ ≥ 2
C
2.25
This means that
1/q− −1
−Δpx z1 2μ ≥ f C g β 2μ
h β 2μ
≥ fz2 hz1 .
2.26
According to 2.24 and 2.26, we can conclude that z1 , z2 is a supersolution for P1 ,
when μ is large enough.
Step 3. We will prove that φ1 ≤ z1 and φ2 ≤ z2 .
Obviously, when μ is large enough, we can easily see that gβ2μ is large enough,
then
f φ2 h φ1 ≤ μ, ∀x ∈ ΩR ,
g φ1 ω φ2 ≤ g β 2μ , ∀x ∈ ΩR .
2.27
8
Journal of Inequalities and Applications
Let us consider
−Δpx μ in ΩR .
2.28
It is easy to see that φ1 is a subsolution of 2.28, when μ is large enough. Obviously,
we can see that z1 is a supersolution of 2.28, and
z1 R φ1 R z1 −R φ1 −R 0.
2.29
According to the comparison principle see 12, we can see that φ1 ≤ z1 .
Let us consider
−Δpx g β 2μ in ΩR .
2.30
It is easy to see that φ2 is a subsolution of 2.30, when μ is large enough. Obviously,
we can see that z2 is a supersolution of 2.30, and
z2 R φ2 R z2 −R φ2 −R 0.
2.31
According to the comparison principle see 12, we can see that φ2 ≤ z2 .
Thus, we can conclude that φ1 ≤ z1 and φ2 ≤ z2 , when μ is sufficiently large. This
completes the proof.
Theorem 2.2. If H1 –H5 hold, then P has infinitely many periodic solutions.
Proof. Let R nT . According to Theorem 2.1, we can conclude that there exists an integer n0
which is large enough such that P1 has a positive solution u#n x, vn# x for any integer n ≥
n0 . Since p and q are even, and f, g, h, and ω are odd, then −u#n −x, −vn# −x is a negative
solution of P1 . We can define a C1 function un x, vn x on −nT, 3nT as
un x vn x ⎧
⎨u#n x,
x ∈ −nT, nT ,
⎩−u# −x − 2nT , x ∈ nT, 3nT ,
n
⎧
⎨vn# x,
x ∈ −nT, nT ,
2.32
⎩−v# −x − 2nT , x ∈ nT, 3nT .
n
We extend un x, vn x as un x, vn x un x m4nT , vn x m4nT , where m is
an integer such that x m4nT ∈ −nT, 3nT . It is easy to see that un , vn ∈ C1 R, un x, vn x
is a solution of P , and the periodic of un x, vn x is 4nT . This completes the proof.
Journal of Inequalities and Applications
9
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.
References
1 E. Acerbi and G. Mingione, “Regularity results for stationary electro-rheological fluids,” Archive for
Rational Mechanics and Analysis, vol. 164, no. 3, pp. 213–259, 2002.
2 E. Acerbi and G. Mingione, “Gradient estimates for the px-Laplacean system,” Journal für die Reine
und Angewandte Mathematik, vol. 584, pp. 117–148, 2005.
3 Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,”
SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406, 2006.
4 X. Fan, Y. Zhao, and D. Zhao, “Compact imbedding theorems with symmetry of Strauss-Lions type
for the space W 1,px Ω,” Journal of Mathematical Analysis and Applications, vol. 255, no. 1, pp. 333–348,
2001.
5 X. Fan, “On the sub-supersolution method for px-Laplacian equations,” Journal of Mathematical
Analysis and Applications, vol. 330, no. 1, pp. 665–682, 2007.
6 X. Fan, Q. Zhang, and D. Zhao, “Eigenvalues of px-Laplacian Dirichlet problem,” Journal of
Mathematical Analysis and Applications, vol. 302, no. 2, pp. 306–317, 2005.
7 A. El Hamidi, “Existence results to elliptic systems with nonstandard growth conditions,” Journal of
Mathematical Analysis and Applications, vol. 300, no. 1, pp. 30–42, 2004.
8 P. Harjulehto, P. Hästö, and V. Latvala, “Harnack’s inequality for p·-harmonic functions with
unbounded exponent p,” Journal of Mathematical Analysis and Applications, vol. 352, no. 1, pp. 345–359,
2009.
9 H. Hudzik, “On generalized Orlicz-Sobolev space,” Functiones et Approximatio Commentarii Mathematici, vol. 4, pp. 37–51, 1976.
10 M. Mihăilescu and V. Rădulescu, “Continuous spectrum for a class of nonhomogeneous differential
operators,” Manuscripta Mathematica, vol. 125, no. 2, pp. 157–167, 2008.
11 M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in
Mathematics, Springer, Berlin, Germany, 2000.
12 Q. Zhang, “A strong maximum principle for differential equations with nonstandard px-growth
conditions,” Journal of Mathematical Analysis and Applications, vol. 312, no. 1, pp. 24–32, 2005.
13 Q. Zhang, “Existence of solutions for px-Laplacian equations with singular coefficients in RN ,”
Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 38–50, 2008.
14 Q. Zhang, “Existence of positive solutions for elliptic systems with nonstandard px-growth
conditions via sub-supersolution method,” Nonlinear Analysis: Theory, Methods & Applications, vol.
67, no. 4, pp. 1055–1067, 2007.
15 Q. Zhang, “Existence and asymptotic behavior of positive solutions to px-Laplacian equations with
singular nonlinearities,” Journal of Inequalities and Applications, vol. 2007, Article ID 19349, 9 pages,
2007.
16 Q. Zhang, “Boundary blow-up solutions to px-Laplacian equations with exponential nonlinearities,” Journal of Inequalities and Applications, vol. 2008, Article ID 279306, 8 pages, 2008.
17 Q. Zhang, X. Liu, and Z. Qiu, “The method of subsuper solutions for weighted pr-Laplacian
equation boundary value problems,” Journal of Inequalities and Applications, vol. 2008, Article ID
621621, 19 pages, 2008.
18 V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory,” Mathematics
of the USSR-Izvestiya, vol. 29, pp. 33–36, 1987.
19 D. D. Hai and R. Shivaji, “An existence result on positive solutions for a class of p-Laplacian systems,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 7, pp. 1007–1010, 2004.
20 C. Chen, “On positive weak solutions for a class of quasilinear elliptic systems,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 62, no. 4, pp. 751–756, 2005.
21 J.-X. Feng and Z.-Q. Han, “Periodic solutions to differential systems with unbounded or periodic
nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1264–1278, 2006.
10
Journal of Inequalities and Applications
22 R. Manásevich and J. Mawhin, “Periodic solutions for nonlinear systems with p-Laplacian-like
operators,” Journal of Differential Equations, vol. 145, no. 2, pp. 367–393, 1998.
23 J. Sun, Y. Ke, C. Jin, and J. Yin, “Existence of positive periodic solutions for the p-Laplacian system,”
Applied Mathematics Letters, vol. 20, no. 6, pp. 696–701, 2007.
24 Y. Tian and W. Ge, “Periodic solutions of non-autonomous second-order systems with a p-Laplacian,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 1, pp. 192–203, 2007.