Infinitely Many Periodic Solutions for Variable Exponent Systems

(1)

Volume 2009, Article ID 714179,10pages doi:10.1155/2009/714179

Research Article

Infinitely Many Periodic Solutions for

Variable Exponent Systems

Xiaoli Guo,

1

_{Mingxin Lu,}

2

_{and Qihu Zhang}

1

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,[email protected]

Received 4 December 2008; Accepted 14 April 2009 Recommended by Marta Garcia-Huidobro

We mainly consider the system−Δpxu fv huinR,−Δqxv gu ωvinR, where

1< px, qx∈C1_R_{are periodic functions, and}₋_Δp

xu−|u|px−2uis calledpx-Laplacian.

We give the existence of infinitely many periodic solutions under some conditions.

Copyrightq2009 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 example1–18 . On the applied background, we refer to1,3,11,18 . In this paper, we mainly consider the existence of infinitely many periodic solutions for the system

P

⎧ ⎨ ⎩

−Δpxufv hu inR, −Δqxvgu ωv inR,

1.1

wherepx, qx∈C1_R_{are functions. The operator}₋_Δ

pxu −|u|px−2uis called one-dimensionalpx-Laplacian. Especially, ifpx ≡ pa constantandqx ≡ qa constant, thenPis the well-known constant exponent system.

(2)

In19 , the authors consider the existence of positive weak solutions for the following constant exponent problems:

I

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

−Δpuλfv, inΩ,

−Δpvλgu, inΩ,

uv0, on∂Ω.

1.2

The first eigenfunction is used to construct the subsolution of constant exponent problems successfully. Under the condition thatλis large enough and

lim

u→∞

fMgu1/p−1

up−1 0, for everyM >0, 1.3

the authors give the existence of positive solutions for problemI.

In20 , the author considers the existence and nonexistence of positive weak solution to the following constant exponent elliptic system:

II

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

−Δpuλuαvγ, inΩ,

−Δqvλuδvβ, inΩ,

uv0, on∂Ω.

1.4

The first eigenfunction is used to construct the subsolution of constant exponent problems successfully.

Because of the nonhomogeneity ofpx-Laplacian,px-Laplacian problems are more complicated than those ofp-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 ofpx-Laplacian, the first eigenfunction cannot be used to construct the subsolution ofpx-Laplacian problems.

There are many papers on the existence of periodic solutions forp-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 problemP.

2. Main Results and Proofs

(3)

Denote ΩR −R, R. Let us consider the existence of positive solutions of the

following:

P1

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

−Δpxufv hu, inΩR,

−Δqxvgu ωv, inΩR,

uv0, on∂ΩR.

2.1

Writezsup_x_∈_Rzx,z−infx∈Rzx, for anyz∈CR. Assume that

H1px, qx∈C1Rsatisfy

1< p−≤p<∞, suppx<∞, 1< q−≤q <∞, supqx<∞. 2.2

H2f,g, h, ω:0,∞ → RareC1, monotone functions such that

lim

t→∞ft t→lim∞gt tlim→∞ht tlim→∞ωt ∞. 2.3

H3For any positive constantM, there are limt→∞fMgt1/q

−₋₁

/tp−−1_0.

H4limt→∞ht/tp−−1limt→∞ωt/tq−−10.

H5f, g, h, andω are odd functions such thatf0 g0 h0 ω0 0, px

andqxare even, andT is a periodic ofpandq, namely,px pxT, qx qxT,for allx∈R.

Note. In14 , the present author discussed the existence of solutions ofP1, under the conditions thatP1is radial,px qx, andhω≡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 problemssee4,14,16 , but the followingTheorem 2.1does not assume any symmetric conditions.

We will establish.

Theorem 2.1. IfH1–H4hold, thenP1possesses a positive solution, whenRis suﬃciently large.

Proof. If we can construct a positive subsolutionφ1, φ2 and supersolutionz1, z2ofP1,

namely,

−Δpxφ1≤fφ2hφ1, −Δqxφ2≤gφ1ωφ2, for a.e. x∈R, −Δpxz1≥fz2 hz1, −Δqxz2≥gz1 ωz2, for a.e.x∈R,

2.4

(4)

Step 1. We will construct a subsolution ofP1. Denfine

φ1x

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

e−k2x−R−₁_, _R−_{a < x}≤_R,

eak2−₁

_R₋_a

x

k2eak2

pR−a−1/pr−1

sinε2r−R−a π

2

1/pr−1 dr,

R−a− π

2ε2

< x≤R−a,

eak2−1

_R₋_a

R−a−π/2ε2

k2eak2

pR−a−1/pr−1

sinε2r−R−a π

2

1/pr−1 dr,

−Ra π

2ε1 < x≤R−a− π

2ε2,

eak1−₁

_x

−Ra

k1eak1

p−Ra−1/pr−1

sinπ

2 −ε1r−−Ra

1/pr−1 dr,

−Ra≤x≤ −Ra π

2ε1,

ek1xR−₁_, −_R≤_{x <}−_R_a.

φ2x

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

e−k4x−R−₁_, _R−_{a < x}≤_R,

eak4−₁

R−a

x

k4eak4qR−a−1/qr−1

sinε4r−R−a π

2

1/qr−1 dr,

R−a− π

2ε4

< x≤R−a,

eak4−1

_R₋_a

R−a−π/2ε4

k4eak4

qR−a−1/qr−1

sinε4r−R−a π

2

1/qr−1 dr,

−Ra π

2ε3 < x≤R−a− π

2ε4,

eak3−₁

_x

−Ra

k3eak3

q−Ra−1/qr−1

sinπ

2 −ε3r−−Ra

1/qr−1 dr,

−Ra≤x≤ −Ra π

2ε3,

ek3xR−₁_, −_R≤_{x <}−_R_a,

2.5

where

amin

infpx−1 4suppx1,

infqx−1 4supqx1

, bminf0, g0, h0, ω0,−1,

εik−p

i e−akip

, i1,2, εik−q

i e−akiq

, i3,4; R >π

εi, i1,2,3,4,

(5)

k1andk2satisfy

eak2−₁

R−a

R−a−π/2ε2

k2eak2

pR−a−1/pr−1

sinε2r−R−a π

2

1/pr−1 dr

eak1−1

₋_R_a_π/₂_ε₁

−Ra

k1eak1

p−Ra−1/pr−1

sinπ

2−ε1r−−Ra

1/pr−1 dr,

2.7

k3andk4satisfy

eak4−₁

_R₋_a

R−a−π/2ε4

k4eak4

qR−a−1/qr−1

sinε4r−R−a π

2

1/qr−1 dr

eak3−₁

₋Raπ/2ε3

−Ra

k3eak3

q−Ra−1/qr−1

sinπ

2 −ε3r−−Ra

1/qr−1 dr,

2.8

then φ1x ∈ C−R, R , and φ2x ∈ C−R, R . It is easy to see that φi ≥ 0 andφi ∈ C1₋_{R, R} _,_i₁_,_{2. Obviously,}_ε

ik−p

i e−akip

is continuous aboutki.

In the following, we will prove thatφ1, φ2is a subsolution forP1. By computation,

−Δpxφ1

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

k2e−k2x−Rpx−1−_k2_p_x−1_p_x_ln_k2−_k2p_x_x−_R_{, R}−_{a < x}≤_R,

ε2k2eak2pR−a−1_cos

ε2x−R−a π

2

, R− π

2ε2

< x < R−a,

0, −Ra π

2ε1

< x < R−a− π

2ε2 ,

ε1k1eak1p−Ra−1

cosπ

2 −ε1x−−Ra , −Ra < x <−Ra,

k1ek1xRpx−1−_k1_p_x−1−_p_x_ln _k1−_k1p_x_x_R_, −_R≤_{x <}−_R_a.

2.9

Ifk2 is suﬃciently large, we have

−Δpxφ1≤ −k2

inf px−1−suppx

ln k2

k2

R−r

≤ −k2a, ∀x∈R−a, R.

2.10

Asais a constant and only depends onpxandqx, whenk2is large enough, we have−k2a <2b. Sinceφ1x≥0 andfhis monotone, we have

(6)

According toH2, whenkiare large enough, we have

feaki−1≥1, geaki−1≥1, heaki−1≥1, ωeaki−1≥1, i1,2,3,4,

2.12

whereki are dependent onf,g,h,ω, p andq, and they are independent on R. Sinceε2 k₂−pe−ak2p_{, when}_x∈_R−_a−_π/₂_{ε, R}−_a_{, we have}

−Δpxφ1ε2

k2eak2

pR−1−1

cosεx−R−a π

2

≤ε2kp 2 eak2p

1. 2.13

Then we have

−Δpxφ1≤1≤f

φ2

hφ1

, R−a− π

2ε2

< x < R−a. 2.14

Obviously

−Δpxφ10≤1≤fφ2hφ1, −Ra₂π_ε

1

< x < R−a− π

2ε2

. 2.15

Whenk2is large enough, from2.7we can see thatk1is large enough. Similar to the

discussion of the above, we can conclude

−Δpxφ1≤1≤fφ2hφ1, −Ra < x <−Ra₂π_ε1, 2.16

−Δpxφ1≤f0 h0≤fφ2hφ1, −R < x <−Ra. 2.17

Sinceφix∈C1−R, R ,combining2.11,2.14,2.15,2.16and2.17, we have

−Δpxφ1≤f

φ2

hφ1

, for a.e.x∈−R, R. 2.18

Similarly, whenk4is large enough, we have

−Δqxφ2 ≤g

φ1

ωφ2

, for a.e.x∈−R, R. 2.19

Thenφ1, φ2is a subsolution ofP1.

Step 2. We will construct a supersolution ofP1.

Letz1be a solution of

−z₁px−2z₁2μ, z1R 0z1−R, 2.20

(7)

Obviously, there existsx0∈ΩRsuch thatz1x

R

x|r−x02μ|

1/pr−1−1₂_μ_r₋_x 0dr.

Note thatx0is dependent onμ. Denoteββ2μ max|x|≤R z1x. It is easy to see that

1

Cμ

1/p−1_≤_β₂_μ_≤_Cμ1/p−−1_, _where_C_≥_{1 is a positive constant}_. _2.21

Let us consider

−pxz12μ inΩR,

−qxz22gβ2μ inΩR,

z1z20 on ∂ΩR.

2.22

Similarly, we have

1

C

gβ2μ1/q−1≤max

|x|≤R z2x≤C

gβ2μ1/q−−1. 2.23

We will prove thatz1, z2is a supersolution forP1. From limt→∞ωt/tq−−10 and

2.23, whenμis large enough, we can easily see that

−Δqxz22g

β2μ≥gz1 ωz2. 2.24

Since limt→∞fCg2t1/q

−₋₁

/tp−−1 _{0 and lim}

t→∞ht/tp

−₋₁

0, whenμis large enough, according to2.21and2.23, we have

2μ≥2

1

Cβ

2μ

p−−1

≥fCgβ2μ1/q−−1 hβ2μ. 2.25

This means that

−Δpxz12μ≥f

Cgβ2μ1/q−−1 hβ2μ≥fz2 hz1. 2.26

According to2.24and2.26, we can conclude thatz1, z2is a supersolution forP1,

whenμis large enough.

Step 3. We will prove thatφ1≤z1andφ2 ≤z2.

Obviously, whenμis large enough, we can easily see thatgβ2μis large enough, then

fφ2

hφ1

≤μ, ∀x∈ΩR,

gφ1ωφ2≤gβ2μ, ∀x∈ΩR.

(8)

Let us consider

−Δpx μinΩR. 2.28

It is easy to see thatφ1is a subsolution of2.28, whenμis large enough. Obviously, we can see thatz1is a supersolution of2.28, and

z1R φ1R z1−R φ1−R 0. 2.29

According to the comparison principlesee12 , we can see thatφ1≤z1.

Let us consider

−Δpx gβ2μinΩR. 2.30

It is easy to see thatφ2is a subsolution of2.30, whenμis large enough. Obviously,

we can see thatz2is a supersolution of2.30, and

z2R φ2R z2−R φ2−R 0. 2.31

According to the comparison principlesee12 , we can see thatφ2≤z2.

Thus, we can conclude thatφ1 ≤ z1 and φ2 ≤ z2, when μis suﬃciently large. This completes the proof.

Theorem 2.2. IfH1–H5hold, thenPhas infinitely many periodic solutions.

Proof. LetRnT. According toTheorem 2.1, we can conclude that there exists an integern0

which is large enough such thatP1has a positive solutionu#

nx, v#nxfor any integern≥ n0. Sincepandqare even, andf, g, h,andωare odd, then−un#−x,−v#n−xis a negative

solution ofP1. We can define aC1functionunx, vnxon−nT,3nT as

unx

⎧ ⎨ ⎩

u#_nx, x∈−nT, nT ,

−u#

n−x−2nT, x∈nT,3nT ,

vnx

⎧ ⎨ ⎩

v#

nx, x∈−nT, nT ,

−v#

n−x−2nT, x∈nT,3nT .

2.32

We extendunx, vnxasunx, vnx unxm4nT, vnxm4nT,wheremis

an integer such thatxm4nT ∈−nT,3nT . It is easy to see thatun, vn∈C1R,unx, vnx

(9)

Acknowledgment

Partly supported by the National Science Foundation of China10701066 & 10671084and China Postdoctoral Science Foundation20070421107and the Natural Science Foundation of Henan Education Committee 2008-755-65 & 2009A120005 and the Natural Science Foundation of Jiangsu Education Committee08KJD110007.

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 thepx-Laplacean system,”Journal f ¨ur 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 spaceW1,px_Ω_,”_{Journal of Mathematical Analysis and Applications, vol. 255, no. 1, pp. 333–348,}

2001.

5 X. Fan, “On the sub-supersolution method forpx-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. H¨ast ¨o, and V. Latvala, “Harnack’s inequality for p·-harmonic functions with

unbounded exponentp,”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 Mathe-matici, vol. 4, pp. 37–51, 1976.

10 M. Mih˘ailescu and V. R˘adulescu, “Continuous spectrum for a class of nonhomogeneous diﬀerential operators,”Manuscripta Mathematica, vol. 125, no. 2, pp. 157–167, 2008.

11 M. R ˚uˇziˇcka,Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 ofLecture Notes in Mathematics, Springer, Berlin, Germany, 2000.

12 Q. Zhang, “A strong maximum principle for diﬀerential equations with nonstandardpx-growth

conditions,”Journal of Mathematical Analysis and Applications, vol. 312, no. 1, pp. 24–32, 2005.

13 Q. Zhang, “Existence of solutions forpx-Laplacian equations with singular coeﬃcients inRN_,” 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 topx-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 topx-Laplacian equations with exponential

nonlineari-ties,”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 ofp-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.

(10)

22 R. Man´asevich 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 thep-Laplacian system,” Applied Mathematics Letters, vol. 20, no. 6, pp. 696–701, 2007.