Journal of Inequalities and Applications Volume 2009, Article ID 191649,12pages doi:10.1155/2009/191649
Research Article
Multiple Solutions for a Class of
p
x
-Laplacian Systems
Yongqiang Fu and Xia Zhang
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Xia Zhang,[email protected]
Received 19 November 2008; Accepted 11 February 2009
Recommended by Ondrej Dosly
We study the multiplicity of solutions for a class of Hamiltonian systems with thepx-Laplacian. Under suitable assumptions, we obtain a sequence of solutions associated with a sequence of positive energies going toward infinity.
Copyrightq2009 Y. Fu and X. Zhang. 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 and Main Results
Since the space Lpx and W1,px were thoroughly studied by Kov´aˇcik and R´akosn´ık 1,
variable exponent Sobolev spaces have been used in the last decades to model various phenomena. In2, R ˚uˇziˇcka presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids.
In recent years, the differential equations and variational problems withpx-growth conditions have been studied extensively; see for example3–6. In7, 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 px-Laplacian systems with “concave and convex nonlinearity”:
−div∇upx−2∇u|u|px−2uHux, u, v, x∈Ω,
−div∇vpx−2∇v|v|px−2v−Hvx, u, v, x∈Ω,
ux vx 0, x∈∂Ω,
1.1
whereΩ⊂RNis a bounded domain,pis continuous onΩand satisfies 1< p
−≤px≤p <
of HamiltoniansHsuch that
Hx, u, v |u|
αx
αx |v|βx
βx Fx, u, v, 1.2
where 1< α− ≤αx≤px, pxβxp∗x.Here we denote
psup
x∈Ωpx, p−x∈infΩpx, 1.3
and denote by px βx the fact that infx∈Ωβx−px > 0.Throughout this paper,
Fx, u, vsatisfies the following conditions:
H1F ∈C1Ω×R2,R.Writingz u, v, Fx,0≡0, Fzx,0≡0;
H2there existpx< q1xp∗x,1< q2−≤q2x< pxsuch that Fux, u, v,Fvx, u, v≤a0
1|u|q1x−1|v|q2x−1, 1.4
wherea0is positive constant;
H3there existμx, νx∈C1Ωwithpxμxp∗x,1< ν
− ≤νx≤px,
andR0>0 such that
1
μxFux, u, vu
1
νxFvx, u, vv≥Fx, u, v>0, 1.5
when|u, v| ≥R0.
As8, Lemma 1.1, from assumptionH3, there existb0, b1>0 such that
Fx, u, v≥b0
|u|μx|v|νx−b
1, 1.6
for anyx, u, v∈Ω×R2.We can also get that there existsb
2>0 such that
1
μxFux, u, vu
1
νxFvx, u, vvb2≥Fx, u, v, 1.7
for anyx, u, v∈Ω×R2.In this paper, we will prove the following result.
Theorem 1.1. Assume that hypotheses (H1)–(H3) are fulfilled. IfFx, zis even inz, then problem 1.1has a sequence of solutions{zn}such that
Izn
Ω
∇unpxunpx
px −
∇vnpx
vnpx
px −H
x, zn
dx−→ ∞, 1.8
2. Preliminaries
First we recall some basic properties of variable exponent spaces LpxΩ and variable
exponent Sobolev spaces W1,pxΩ,where Ω ⊂ RN is a domain. For a deeper treatment
on these spaces, we refer to1,9–11.
LetPΩbe the set of all Lebesgue measurable functionsp:Ω → 1,∞and
|u|pxinf
λ >0 :
Ω
uλpxdx≤1
. 2.1
The variable exponent spaceLpxΩis the class of all functionsusuch that
Ω|ux|pxdx <
∞.Under the assumption thatp < ∞, LpxΩis a Banach space equipped with the norm
2.1.
The variable exponent Sobolev space W1,pxΩ is the class of all functions u ∈
LpxΩsuch that|∇u| ∈LpxΩand it can be equipped with the norm
u1,px|u|px|∇u|px. 2.2
Foru∈W1,pxΩ,if we define
|||u|||inf
λ >0 :
Ω
|u|px|∇u|px
λpx dx≤1
, 2.3
then|||u|||andu1,pxare equivalent norms onW1,pxΩ.
By W01,pxΩ we denote the subspace ofW1,pxΩ which is the closure ofC∞0 Ω
with respect to the norm2.2and denote the dual space ofW01,pxΩbyW−1,pxΩ.We
know that ifΩ ⊂ RN is a bounded domain,||u||
1,px and |∇u|px are equivalent norms on
W01,pxΩ.
Under the condition 1< p− ≤ p <∞, W01,pxΩis a separable and reflexive Banach
space, then there exist{en}n1∞ ⊂W01,pxΩand{fm}m∞1⊂W−1,p
x
Ωsuch that
fmen
1 ifnm,
0 ifn /m, W01,pxΩ spanei:i1, . . . , n, . . .
,
W−1,pxΩ spanfj:j 1, . . . , m, . . .
.
2.4
In the following, we will denote thatEE1⊕E2,where
For any z ∈ E, define the norm||z|| ||u, v|| |||u||||||v|||. For anyn ∈ N,set e1
n 0, en, e2
n en,0and
Xnspan
e11, . . . , en1
⊕E2, XnE1⊕spane12, . . . , e2n
, 2.6
denote the complement ofXninEbyXn⊥span{e2
n1, en22, . . .}.
3. The Proof of Theorem
1.1
Definition 3.1. We say thatz0 u0, v0∈Eis a weak solution of problem1.1, that is,
Ω
∇u0px−2∇u0∇uu0px−2u0u−∇v0px−2∇v0∇v
−v0px−2v0v−Hu
x, u0, v0
u−Hv
x, u0, v0
vdx0, ∀z∈E.
3.1
In this section, we denote thatVm span{ei :i 1, . . . , m},for anym ∈N, andci is
positive constant, for anyi0,1,2. . . .
Lemma 3.2. AnyPSsequence{zn} ⊂ E, that is,|Izn| ≤ cand Izn → 0,as n → ∞,is bounded.
Proof. Lets > 0 be sufficiently small such thatl1 infx∈Ω1/px −1s/μx > 0, l2
infx∈Ω1s/νx−1/px>0, l3supx∈Ω1/αx−1s/μx>0, l4supx∈Ω1
s/νx− 1/ βx >0.
Let{zn} ⊂Ebe such that|Izn| ≤c and Izn → 0,asn → ∞.We get
Izn
−
Izn
,
1s μxun,
1s νxvn
Ω 1
px −
1s μx
∇unpxunpx
1sun
μx2 ∇un
px−2∇u
n∇μ
1s νx −
1
px
∇vnpxvnpx
− 1svn
νx2
∇vnpx−2∇v
n∇ν
1s
μxFu
x, un, vn
un
1s νxFv
x, un, vn
vn−F
x, un, vn
1s μx −
1
αx
unαx
1s νx −
1
βx
vnβx
dx ≥ Ω
l1∇unpxl2∇vnpxsF
x, un, vn
−l3unαxl4vnβx
1sun
μx2
∇unpx−2∇u
n∇μ− 1svn
νx2
∇vnpx−2∇v
n∇ν−1sb2
dx.
Asμx, νx∈C1Ω,by the Young inequality, we can get that for anyε
1, ε2∈0,1,
1sun
μx2 ∇un
px−2∇u
n∇μ
≤c0∇unpx−1un
≤c0
ε1px−1
px ∇un
px ε11−px
px un
px
≤c0
ε1∇unpxε11−punpx
,
1svn
νx2
∇vnpx−2∇v
n∇ν
≤c1
ε2∇vnpxε21−pvnpx
.
3.3
Letε1, ε2be sufficiently small such that
c0ε1≤ l1
2 , c1ε2≤
l2
2 , 3.4
then
Izn
−
Izn
,
1s μxun,
1s νxvn
≥
Ω
l1
2 ∇un
px l2
2 ∇vn
px
sb0unμxb0vnνx−b1
−l3unαxc0ε11−punpx
l4vnβx−c1ε12−pvnpx
−1sb2
dx.
3.5
Note thatαx≤pxμx, pxβx,by the Young inequality, for anyε3, ε4, ε5∈0,1,
we get
unαx≤ ε3αxunμx
μx
μx−αx μx ε
αx/αx−μx 3
≤ε3unμxε−α3 /μ−α−,
unpx≤ ε4px
μx un
μx μx−px
μx ε
px/px−μx 4
≤ε4unμxε−p4 /μ−p−,
vnpx≤ ε5px
βx vn
βx βx−px
βx ε
px/px−βx 5
≤ε5vnβxε−p5 /β−p−.
Letε3, ε4, ε5 be sufficiently small such thatl3ε3c0ε11−pε4 ≤sb0 andc1ε12−pε5 ≤ l4,then we
get
Izn
−
Izn
,
1s μxun,
1s νxvn
≥ Ω l1
2 ∇un
px l2
2 ∇vn
px−
c2
dx. 3.7
Note that
Izn
,
1s μxun,
1s νxvn
≤Izn·1s μxun
1s
νxvn
≤c3I
zn·
∇
1s μxun
px ∇
1s νxvn
px
≤c4I
zn·
∇unpx∇vnpx
,
3.8
and forn∈Nbeing large enough, we have
c4I
zn≤min
l1 4 , l2 4
. 3.9
It is easy to know that if|∇un|px≥1 and|∇vn|px≥1,
∇un
px ≤
Ω
∇unpx
dx, ∇vnpx≤
Ω
∇vnpx
dx, 3.10
thus we get
Izn
≥ Ω l1
4 ∇un
px l2
4 ∇vn
px−
c2
dx, 3.11
then|∇un|px,|∇vn|pxare bounded. Similarly, if|∇un|px < 1 or|∇vn|px < 1,we can also
get that|∇un|px,|∇vn|pxare bounded. It is immediate to get that{zn}is bounded inE.
Lemma 3.3. AnyPSsequence contains a convergent subsequence.
Proof. Let{zn} ⊂ Ebe aPSsequence. ByLemma 3.2, we obtain that{zn}is bounded inE.
existsz ∈Esuch thatzn → zweakly inE.Then we can getun → uweakly inW01,pxΩ.
Note that
Izn
−Iz,un−u,0
Ω
∇unpx−2∇un−∇upx−2∇u
∇un−u
unpx−2un− |u|px−2uun−u
−unαx−2un− |u|αx−2u
un−u
−Fu
x, un, vn
−Fux, u, vun−u
dx.
3.12
It is easy to get that
Izn
−Iz,un−u,0
−→0,
ΩFux, u, v
un−u
dx−→0, 3.13
andun → uinLpxΩ, un → uinLαxΩ,asn → ∞.Then
Ω
unpx−2un− |u|px−2uun−u
dx−→0,
Ω
unαx−2un− |u|αx−2u
un−u
dx−→0,
3.14
asn → ∞.By conditionH2, we obtain
Ω Fu
x, un, vn
un−udx
≤
Ωa0
1unq1x−1vnq2x−1
un−udx
≤a1
un−u1unq1x−1
q1x·un−uq1x
vnq2x−1 q2x
·un−uq2x
.
3.15
It is immediate to get that |un −u|1 → 0,||un|q1x−1|q1x,||vn|q2x−1|q2x are bounded and
|un−u|q1x → 0,|un−u|q2x → 0,then we get
ΩFu
x, un, vn
un−u
dx−→0,
Ω
∇unpx−2∇un−∇upx−2∇u
∇un−u
dx−→0,
3.16
asn → ∞.Similar to3,4, Theorem 3.1, we divideΩinto two parts:
OnΩ1,we have
Ω1
∇un− ∇upx
dx
≤c5
Ω1
∇unpx−2∇un−∇upx−2∇u∇un− ∇u
px/2
×∇unpx∇upx
2−px/2
dx
≤c6
∇unpx−2∇un−∇upx−2∇u∇un− ∇u
px/2
2/ px,Ω1
×∇unpx∇upx
2−px/2
2/2−px,Ω1
,
3.18
thenΩ
1|∇un− ∇u|
pxdx → 0.OnΩ
2,we have
Ω2
|∇un− ∇u|pxdx≤c7
Ω2
|∇un|px−2∇un− |∇u|px−2∇u∇un− ∇udx−→0. 3.19
Thus we getΩ|∇un− ∇u|pxdx → 0.Thenun → uin W01,pxΩ,asn → ∞.Similarly,
vn → vinW01,pxΩ.
Lemma 3.4. There existsRm>0such thatIz≤0for allz∈Xmwith||z|| ≥Rm.
Proof. For anyz u, v∈Xm, u∈Vm,we have
Iz≤
Ω
∇upx|u|px
px −
∇vpx|v|px
px −Fx, u, v dx
≤
Ω
∇upx|u|px
p− −
∇vpx|v|px
p −b0|u|
μxb
1 dx.
3.20
In the following, we will considerΩ|∇u|px|u|px/p−−b0|u|μxdx.
iIf|||u||| ≤1.We have
Ω
∇upx|u|px
p− −b0|u|
μx dx≤ 1
p− . 3.21
iiIf|||u|||>1.Note thatμ, p ∈CΩ, pxμx.For anyx∈Ω,there existsQx
which is an open subset ofΩsuch that
px sup
then{Qx}x∈Ωis an open covering ofΩ.AsΩis compact, we can pick a finite subcovering
{Qx}ni1forΩ.Thus there exists a sequence of open set{Ωi}ni1such thatΩ ni1Ωiand
pi sup
x∈Ωipx< μi−
inf
x∈Ωiμx, 3.23
fori1, . . . , n.Denote thatri|||u|||Ωi,then we have
Ω
∇upx|u|px
p− −b0|u|
μx dx
n i1
Ωi
∇upx|u|px
p− −b0|u|
μx dx
ri>1
Ωi
∇upx|u|px
p− −b0|u|
μx dx
ri≤1
Ωi
∇upx|u|px
p− −b0|u|
μx dx
≤
ri>1
|||u|||pi
Ωi
p− −b0kmi|||u||| μi−
Ωi pn− ,
3.24
where kmi infu∈Vm|Ωi,|||u|||Ωi1
Ωi|u|μxdx.As Vm|Ωi is a finite dimensional space, we have
kmi >0,fori1, . . . , n.
We denote bysi the maximum of polynomial tpi/p− −b0kmitμi− on 0,∞,fori
1, . . . , n.Then there existst0>1 such that
tpi
p− −b0kmit μi−c
8≤0, 3.25
fort > t0andi1, . . . , n,wherec8 ni1sin/p−b1measΩ.
LetRm max{2,2pc8 1/ p−1/p−,2nt0}.If ||z|| ≥ Rm,we get |||u||| ≥ Rm/2 or
|||v||| ≥Rm/2.
iIf|||u||| ≥Rm/2,|||u||| ≥nt0 > 1.It is easy to verify that there exists at leasti0such
that|||u|||Ωi
0 ≥t0>1,thus
Iz≤ |||u|||
pi0
Ωi0
p− −b0kmi0|||u||| μi0−
iiIf|||v||| ≥Rm/2,|||v||| ≥pc81/p−1/p−.We obtain
Iz≤c8
1
p− −
|||v|||p−
p ≤0. 3.27
Now we get the result.
Lemma 3.5. There existrm>0andam → ∞m → ∞such thatIz≥am,for anyz∈Xm−1⊥ with||z||rm.
Proof. Forz u, v∈Xm−1⊥, v0.By conditionH2, there existsc
9>0 such that Fx, u,0≤c9|u|q1xc
9. 3.28
Let||z|| ≥1,we get
Iz
Ω
∇upx|u|px
px − |u|αx
αx −Fx, u,0 dx
≥
Ω
∇upx|u|px
p −
|u|αx
α− −c9|u| q1x−c
9 dx
≥
Ω
∇upx|u|px
p −c10|u|
q1x dx−c
11.
3.29
Denote that
θm sup u∈V⊥
m
|||u|||≤1
Ω|u|
q1xdx, 3.30
thus
Iz≥ |||u|||
p−
p −c10θmu
q1−c
11. 3.31
Let
rmmax
1,
p−
c10pq1θm
1/q1−p−
,
2
c11pq1
q1−p−
1/p−
By5, Lemma 3.3, we get thatθm → 0,asm → ∞,then
Iz≥rmp−
q1−p−
pq1 −c11
am,
3.33
whenmis sufficiently large and||z||rm.It is easy to get thatam → ∞,asm → ∞.
Lemma 3.6. I is bounded from above on any bounded set ofXm.
Proof. Forz u, v∈Xm.We get
Iz≤
Ω
∇upx|u|px
px −Fx, u, v dx. 3.34
By conditionsH2and H3, we know that if|u, v| ≥ R0, Fx, u, v ≥ 0 and if |u, v| <
R0,|Fx, u, v| ≤c0.Then
Iz≤
Ω
∇upx|u|px
px c12 dx, 3.35
and it is easy to get the result.
Proof ofTheorem 1.1. By Lemmas3.2–3.6above, and7, Proposition 2.1 and Remark 2.1, we know that the functionalI has a sequence of critical valuesck → ∞,ask → ∞.Now we
complete the proof.
Acknowledgments
This work is supported by Science Research Foundation in Harbin Institute of Technology
HITC200702and The Natural Science Foundation of Heilongjiang ProvinceA2007-04.
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