Establishing the suitable conditions on the nonlinearity, we proved the existence of nontrivial solutions.. Using a variational approach, the authors prove the existence of nontrivial so
Trang 1Volume 2008, Article ID 612938, 16 pages
doi:10.1155/2008/612938
Research Article
Existence of Solutions for a Class of Elliptic
S Ogras, R A Mashiyev, M Avci, and Z Yucedag
Department of Mathematics, Dicle University, 21280 Diyarbakir, Turkey
Correspondence should be addressed to R A Mashiyev,mrabil@dicle.edu.tr
Received 4 April 2008; Accepted 17 July 2008
Recommended by M Garcia Huidobro
In view of variational approach, we discuss a nonlinear elliptic system involving the
px-Laplacian Establishing the suitable conditions on the nonlinearity, we proved the existence of nontrivial solutions
Copyrightq 2008 S Ogras 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 paper concerns the existence of nontrivial solutions for the following nonlinear elliptic system:
−Δpx u ∂F
∂u x, u, υ in R N ,
−Δqx υ ∂F
∂υ x, u, υ in R N ,
P,Q
where px and qx are two functions such that 1 < px, qx < N N ≥ 2, for every
x ∈ R N However, F ∈ C1RN × R2 and Δpx is the px-Laplacian operator defined by
Δpx u div|∇u| px−2 ∇u Using a variational approach, the authors prove the existence of
nontrivial solutions
Over the last decades, the variable exponent Lebesgue space L px and Sobolev space
W 1,px 1 5 have been a subject of active research stimulated mainly by the development
of the studies of problems in elasticity, electrorheological fluids, image processing, flow in porous media, calculus of variations, and differential equations with px-growth conditions
6 13
Among these problems, the study of px-Laplacian problems via variational methods
is an interesting topic A lot of researchers have devoted their work to this area 14–22
Trang 2The operatorΔpx u : div|∇u| px−2 ∇u is called px-Laplacian, where p is a continuous nonconstant function In particular, if px ≡ p constant, it is the well-known p-Laplacian operator However, the px-Laplace operator possesses more complicated nonlinearity than
p-Laplace operator due to the fact that Δ px is not homogeneous This fact implies some difficulties, as for example, we cannot use the Lagrange multiplier theorem and Morse theorem in a lot of problems involving this operator
In literature, elliptic systems with standard and nonstandard growth conditions have been studied by many authors23–28, where the nonlinear function F have different and
mixed growth conditions and assumptions in each paper
In29, the authors show the existence of nontrivial solutions for the following
p-Laplacian problem:
−Δp u ∂F
∂u x, u, υ in R N ,
−Δq υ ∂F
∂υ x, u, υ in R N ,
1.1
where F ∈ C1RN × R2 yields some mixed growth conditions and the primitive F being
intimately connected with the first eigenvalue of an appropriate system Using a weak version of the Palais-Smale condition, that is, Cerami condition, they apply the mountain pass theorem to get the nontrivial solutions of the the system
In30, the author obtains the existence and multiplicity of solutions for the following problem:
− div|∇u| px−2 ∇u ∂F
∂u x, u, υ in Ω,
− div|∇υ| qx−2 ∇υ ∂F
∂υ x, u, υ in Ω,
u 0, υ 0 on Ω,
1.2
where Ω ⊂ RN is a bounded domain with a smooth boundary ∂Ω, N ≥ 2, p, q ∈
CΩ2, px > 1, qx > 1, for every x ∈ Ω The function F is assumed to be continuous
in x ∈ Ω and of class C1 in u, υ ∈ R Introducing some natural growth hypotheses on the
right-hand side of the system which will ensure the mountain pass geometry and Palais-Smale condition for the corresponding Euler-Lagrange functional of the system, the author
limits himself to the subcritical case for function F to obtain the existence and multiplicity
results
In the paper31, Xu and An deal with the following problem:
− div|∇u| px−2 ∇u |u| px−2
u ∂F
∂u x, u, υ in R N ,
− div|∇υ| qx−2 ∇υ |υ| qx−2
υ ∂F
∂υ x, u, υ in R N ,
u, υ ∈ W 1,pxRN × W 1,qxRN ,
1.3
where N ≥ 2, px, qx are functions on R N The function F is assumed to satisfy Caratheodory conditions and to be L∞in x ∈ R N and C1 in u, υ ∈ R By the critical point
Trang 3theory, the authors use the two basic results on the existence of solutions of the system; these
results correspond to the sublinear and superlinear cases for p 2, respectively.
Inspired by the above-mentioned papers, we concern the existence of nontrivial solutions of problemP,Q We know that in the study of px-Laplace equations in RN, the main difficulty arises from the lack of compactness So, establishing some growth conditions
on the right-hand side of the system which will ensure the mountain pass geometry
and Cerami condition for the corresponding Euler-Lagrange functional J and applying a subcritical case for function F, we will overcome this difficulty.
2 Notations and preliminaries
We will investigate our problemP,Q in the variable exponent Sobolev space W1,px
0 RN,
so we need to recall some theories and basic properties on spaces L pxRN and W 1,pxRN .
Set
C RN
h ∈ CR N : inf
x∈R N hx > 1
For every h ∈ C RN, denote
h− : inf
x∈R N hx, h : sup
x∈R N
Let us define byURN the set of all measurable real-valued functions defined on RN
For p ∈ C RN, we denote the variable exponent Lebesgue space by
L pxRN
u ∈ UR N :
RN |ux| px
dx < ∞
which is equipped with the norm, so-called Luxemburg norm1,3,4:
|u| px: |u|L pxRN inf
λ > 0 :
RN
ux λ px dx ≤ 1
andL pxRN , |·| L pxRN becomes a Banach space, we call it generalized Lebesgue space
Define the variable exponent Sobolev space W 1,pxRN by
W 1,pxRN {u ∈ L pxRN : |∇u| ∈ L pxRN }, 2.5 and it can be equipped with the norm
u 1,px: u W 1,px |u| px |∇u| px ∀u ∈ W 1,pxRN . 2.6
The space W01,pxRN is denoted by the closure of C∞
0 RN in W 1,pxRN and it is
equipped with the norm for all u ∈ W01,pxRN:
u px |∇u| px ∀u ∈ W 1,px
0 RN . 2.7
If p− > 1, then the spaces L pxRN , W 1,pxRN , and W 1,px
0 RN are separable and reflexive Banach spaces
Trang 4Proposition 2.1 see 1,3,4 The conjugate space of L pxRN is L p xRN , where 1/p x 1/px 1 For any u ∈ L pxRN and v ∈ L p xRN , we have
RN uv dx
≤p1− 1
p −
|u| px |v| p x ≤ 2|u| px |v| p x 2.8
Proposition 2.2 see 1,3,4 Denote px u RN |ux| px dx for all u ∈ L pxRN , one has
min
|u| p− px , |u| p px ≤ px u ≤ max |u| p−
px , |u| p px . 2.9
Proposition 2.3 see 1 Let px and qx be measurable functions such that px ∈ L∞RN
and 1 ≤ pxqx ≤ ∞, for a.e x ∈ R N Let u ∈ L qxRN , u / 0 Then,
|u| pxqx ≤ 1 ⇒ |u| p
pxqx≤|u| px
qx ≤ |u| p−
pxqx ,
|u| pxqx ≥ 1 ⇒ |u| p−
pxqx≤|u| px
qx ≤ |u| p
pxqx
2.10
In particular, if px p is constant, then
||u| p|qx |u| p pqx 2.11
Proposition 2.4 see 3,4 If u, u n ∈ L pxRN , n 1, 2, , then the following statements are
equivalent to each other:
1 limn→∞ |u n − u| px 0,
2 limn→∞ u n − u 0,
3 u n → u in measure in R N and lim n→∞ u n u.
Definition 2.5 1 < px < N and for all x ∈ R N, let define
p∗x
⎧
⎪
⎪
Npx
N − px if px < N,
∞ if px ≥ N, where p∗x is the so-called critical Sobolev exponent of px.
Proposition 2.6 see 1,32 Let px ∈ C 0,1
RN , that is, Lipschitz-continuous function defined
onRN , then there exists a positive constant c such that
|u| p∗x ≤ c u px , 2.12
for all u ∈ W01,pxRN .
In the following discussions, we will use the product space
W px,qx: W1,px
0 RN × W 1,qx
0 RN , 2.13
Trang 5which is equipped with the norm
u, υ px,qx: max u px υ qx ∀u, υ ∈ W px,qx , 2.14 where u px resp., u qx is the norm of W 1,px
0 RN resp., W 1,qx
0 RN The space
W px,qx∗ denotes the dual space of W px,qxand equipped with the norm · ∗,px,qx Thus,
J u, υ ∗,px,qx D1 Ju, υ ∗,px D2 Ju, υ ∗,qx , 2.15
where W −1,p xRN resp., W −1,q xRN is the dual space of W 1,px
0 RN resp.,
W01,qxRN, and · ∗,pxresp., · ∗,qx is its norm
Foru, υ and ϕ, ψ in W px,qx, let
Fu, υ
RN Fx, ux, υxdx. 2.16 Then,
F u, υϕ, ψ D1Fu, υϕ D2Fu, υψ, 2.17 where
D1Fu, υϕ
RN
∂F
∂u x, u, υϕ dx,
D2Fu, υψ
RN
∂F
∂υ x, u, υψ dx.
2.18
The Euler-Lagrange functional associated toP,Q is defined by
Ju, υ
RN
1
px |∇u| px
dx
RN
1
qx |∇υ| qx
dx − u, υ. 2.19
It is easy to verify that J ∈ C1W px,qx , R and that
J u, υϕ, ψ D1 Ju, υϕ D2Ju, υψ, 2.20 where
D1Ju, υϕ
RN |∇u| px−2 ∇u∇ϕ dx − D1Fu, υϕ,
D2Ju, υψ
RN
|∇υ| qx−2 ∇υ∇ψ dx − D2Fu, υψ.
2.21
Definition 2.7 u, υ is called a weak solution of the system P,Q if
RN
|∇u| px−2 ∇u∇ϕ dx
RN
|∇υ| qx−2 ∇υ∇ψ dx
RN
∂F
∂u x, u, υϕ dx
RN
∂F
∂υ x, u, υψ dx,
2.22 for allϕ, ψ ∈ W px,qx
Trang 6Definition 2.8 We say that J satisfies the Cerami condition C if every sequence ω n ∈
W px,qxsuch that
|Jω n | ≤ c, 1 ω n J ω n −→ 0 2.23
contains a convergent subsequence in the norm of W px,qx
In this paper, we will use the following assumptions:
F1 F ∈ C1RN× R2, R and Fx, 0, 0 0;
F2 for all u, υ ∈ R2and for a.e x ∈ R N
∂F ∂u x, u, υ
≤ a1x|u, υ|p−−1 a2x|u, υ| p −1,
∂F ∂υ x, u, υ
≤ b1x|u, υ|q−−1 b2x|u, υ| q −1,
2.24
where
1 < p−, q−≤ p , q < p∗−, q∗−,
a i ∈ L δxRN ∩ L βxRN , b i ∈ L γxRN ∩ L βxRN , i 1, 2,
δx px
px − 1 , γx
qx
qx − 1 , px p∗xpx
p∗x − px ,
qx q∗xqx
q∗x − qx , βx
p∗xq∗x
p∗xq∗x − p∗x q∗x;
2.25
F3 u, υ·∇Fx, u, υ − Fx, u, υ ≤ 0 for all x, u, υ ∈ R N× R2\ {0, 0}, where ∇F
∂F/∂u, ∂F/∂υ;
F4 suppose there exist two positive and bounded functions a ∈ L N/pxRN and b ∈
L N/qxRN such that
lim
|u,υ|→0sup pxqx|Fx, u, υ|
qxax|u| px pxbx|υ| qx
< λ1< lim
|u,υ|→ ∞inf pxqx|Fx, u, υ|
qxax|u| px pxbx|υ| qx
2.26
Let λ1denote the first eigenvalue of the nonlinear eigenvalue problem inRN:
−Δpx u λax|u| px−2 u inRN ,
−Δqx υ λbx|υ| qx−2 υ inRN 2.27
It is useful to recall the variational characterization:
λ1 inf
RN 1/px|∇u| px 1/qx|∇υ| qx dx
RN ax/px|u| px bx/qx|υ| qx dx :u, υ ∈ W px,qx \ {0, 0}
.
2.28
We will assume that λ1is a positive real number for allu, υ ∈ W px,qx \ {0, 0} For
more details about the eigenvalue problems, we refer the reader to17
Trang 73 The main results
We will use the mountain pass theorem together with the following lemmas to get our main results
Lemma 3.1 Under the assumptions F1 and F2, the functional F is well defined, and it is of class
C1on W px,qx Moreover, its derivative is
F u, υω, z
RN
∂F
∂u x, u, υω dx
RN
∂F
∂υ x, u, υz dx ∀u, υ, ω, z ∈ W px,qx
3.1
Proof For all pair of real functions u, υ ∈ W px,qx, under the assumptionsF1 and F2,
we can write
Fx, u, υ
u
0
∂F
∂s x, s, υds Fx, 0, υ
u 0
∂F
∂s x, s, υds
υ 0
∂F
∂s x, 0, sds Fx, 0, 0,
Fx, u, υ ≤ c1a1x|u|p− |υ| p−−1|u| a2x|u| p |υ| p −1|u| b1x|υ| q− b2x|υ| q .
3.2 Then,
RN Fx, u, υdx ≤ c2
RN a1x|u|p−
dx
RN a1x|υ|p−−1|u|dx
RN a2x|u|p
dx
RN a2x|υ|p −1|u|dx
RN b1x|υ|q−
dx
RN b2x|υ|q
dx
,
3.3
if we consider the fact that
W01,pxRN → L p−pxRN ⇒ ||u| p−|px |u| p−
p−px ≤ c u p−
px for p−> 1, 3.4
and if we apply Propositions 2.1, 2.3, and 2.6 and take a i ∈ L δxRN ∩ L βxRN , b i ∈
L γxRN , then we have
RN Fx, u, υdx ≤ 2c1
|a1| δx ||u| p−|px |a1| βx ||υ| p−−1|q∗x |u| p∗x |a2| δx ||u| p |px
|a2| βx ||υ| p −1|q∗x |u| p∗x |b1| γx ||υ| q−|qx |b2| γx ||υ| q |qx
2c1|a1| δx |u| p−
p−px |a1| βx |υ| p−−1
p−−1q∗x |u| p∗x |a2| δx |u| p
p px
|a2| βx |υ| p −1
p −1q∗x |u| p∗x |b1| γx |υ| q−
q−qx |b2| γx |υ| q
q qx
≤ c3|a1| δx u p−
px |a1| βx υ p−−1
qx u px |a2| δx u p
px
|a2| βx υ p −1
qx u px |b1| γx υ q−
qx |b2| γx υ q
qx
< ∞.
3.5
Trang 8Hence,F is well defined Moreover, one can see easily that F is also well defined on W px,qx
Indeed, usingF2 for all ω, z ∈ W px,qx, we have
F u, υω, z
RN
∂F
∂u x, u, υω dx
RN
∂F
∂υ x, u, υz dx,
F u, υω, z ≤
RN
a1x|u, υ| p−−1 a2x|u, υ| p −1|ω|dx
RN b1x|u, υ| q−−1 b2x|u, υ| q −1|z|dx
≤
RN a1x|u|p−−1|ω|dx
RN a1x|υ|p−−1|ω|dx
RN
a2x|u|p −1|ω|dx
RN
a2x|υ|p −1|ω|dx
RN b1x|u|q−−1|z|dx
RN b1x|υ|q−−1|z|dx
RN
b2x|u|q −1|z|dx
RN
b2x|υ|q −1|z|dx,
3.6
and applying Propositions2.1,2.3, and2.6and considering the conditions px > px and
qx > qx, it follows that
RN
∂F
∂u x, u, υω dx ≤ 2|a1| δx ||u| p−−1|p∗x |ω| px |a1| βx ||υ| p−−1|q∗x |ω| p∗x
|a2| δx ||u| p −1|p∗x |ω| px |a2| βx ||υ| p −1|q∗x |ω| p∗x
≤ 2|a1| δx |u| p−−1
p−−1p∗x |ω| px |a1| βx |υ| p−−1
p−−1q∗x |ω| p∗x
|a2| δx |u| p −1
p −1p∗x |ω| px |a2| βx |υ| p −1
p −1q∗x |ω| p∗x
≤ c4|a1| δx u p px−−1 |a1| βx υ p qx−−1 |a2| δx u p px −1 |a2| β υ p qx −1 ω px
< ∞,
3.7 and similarly
RN
∂F
∂υ x, u, υz dx
≤ c5|b1| βx u q−−1
px |b1| γ x υ q−−1
qx |b2| βx u q −1
px |b2| γx υ q −1
qx
z qx < ∞.
3.8
Now let us show thatF is differentiable in sense of Fr´echet, that is, for fixed u, υ ∈
W px,qx and given ε > 0, there must be a δ ε, u, υ > 0 such that
|Fu ω, υ z − Fu, υ − F u, υω, z| ≤ ε ω px z qx , 3.9 for allω, z ∈ W px,qxwith ω px z qx ≤ δ.
Trang 9Let B r be the ball of radius r which is centered at the origin of R N and denote B r
RN − B r Moreover, let us define the functional F r on W01,px B r × W01,qx B r as follows:
Fr u, υ
Br
Fx, ux, υxdx. 3.10
If we considerF1 and F2, it is easy to see that F r ∈ C1W 1,px
0 B r × W 1,qx
0 B r , and in
addition for allω, z ∈ W 1,px
0 B r × W 1,qx
0 B r , we have
F
r u, υω, z
Br
∂F
∂u x, u, υω dx
Br
∂F
∂υ x, u, υz dx. 3.11 Also as we know, the operator F
r : W px,qx → W∗
px,qx is compact 3 Then, for all
u, υ, ω, z ∈ W px,qx , we can write
Fu ω,υ z − Fu,υ − F u, υω, z
≤Fr u ω, υ z − F r u, υ − F
r u, υω, z
B r
Fx, u ω, υ z − Fx, u, υ − ∂F
∂u x, u, υω − ∂F
∂υ x, u, υzdx
.
3.12
By virtue of the mean-value theorem, there exist ζ1 , ζ2∈ 0, 1 such that
Fx, u ω, υ z − Fx, u, υ ∂F
∂u x, u ζ1 ω, υω ∂F
∂υ x, u, υ ζ2 zz. 3.13 Using the conditionF2, we have
B r
∂F
∂u x, u ζ1 ω, υω ∂F
∂υ x, u, υ ζ2 zz − ∂F
∂u x, u, υω − ∂F
∂υ x, u, υzdx
≤
B r
a1x|u ζ1 ω| p−−1− |u| p−−1
a2x|u ζ1 ω| p −1− |u| p −1
|ω|dx
B r
b1x|υ ζ2 z| q−−1− |υ| q−−1
b2x|υ ζ2 z| q −1− |υ| q −1
|z|dx
.
3.14
By help of the elementary inequality|a b| s≤ 2s−1 |a| s |b| s for a, b ∈ R N, we can write
≤ 2p−−1− 1
B r
a1x|u|p−−1|ω|dx ζ12 p−−1
B r
a1x|ω|p−−1|ω|dx
2p −1− 1
B r
a2x|u|p −1|ω|dx ζ12 p −1
B r
a2x|ω|p −1|ω|dx
2q−−1− 1
B r
b1x|υ|q−−1|z|dx ζ22 q−−1
B r
b1x|z|q−−1|z|dx
2q −1− 1
B
b2x|υ|q −1|z|dx ζ22 q −1
B
b2x|z|q −1|z|dx,
3.15
Trang 10applying Propositions2.1,2.3, and2.6, then we have
≤ c6|a1| δx ||u| p−−1|p∗x |ω| px |a1| δx ||ω| p−−1|p∗x |ω| px
|a2| δx ||u| p −1|p∗x |ω| px |a2| δx ||ω| p −1|p∗x |ω| px
|b1| γx ||υ| q−−1|q∗x |z| qx |b1| γx ||z| q−−1|q∗x |z| qx
|b2| γx ||υ| q −1|q∗x |z| qx |b2| γx ||z| q −1|q∗x |z| qx,
≤ c7|a1| δx u p−−1
px |a1| δx ω p−−1
px
|a2| δx u p −1
px |a2| δx ω p −1
px
ω px
|b1| γx υ q−−1
qx |b1| γx z q−−1
qx
|b2| γx υ q −1
qx |b2| γx z q −1
qx
z qx ,
3.16
and by the fact that
|a i|L δx B
r−→ 0,
|b i|L γx B
for i 1, 2, as r → ∞, and for r sufficiently large, it follows that
B r
Fx, u ω, υ z − Fx, u, υ − ∂F
∂u x, u, υω − ∂F
∂υ x, u, υ
zdx
≤ ε ω px z qx .
3.18
It remains only to show that F
is continuous on W px,qx Let u n , υ n , u, υ ∈
W px,qxsuch thatu n , υ n → u, υ Then, for ω, z ∈ W px,qx, we have
|F u n , υ n ω, z − F u, υω, z| ≤ |F
r u n , υ n ω, z − F
r u, υω, z|
B r
∂F
∂u x, u n , υ n ∂F
∂u x, u, υ
ω dx
B r
∂F
∂υ x, u n , υ n ∂F
∂υ x, u, υ
z dx
,
3.19
then byF2, we can write
B r
a1x|un|p−−1 |u| p−−1 |υ n|p−−1 |υ| p−−1|ω|dx 3.20
B r
a2x|un|p −1 |u| p −1 |υ n|p −1 |υ| p −1|ω|dx I1
B r
b1x|un|q−−1 |u| q−−1 |υ n|q−−1 |υ| q−−1|z|dx I2
B
b2x|un|q −1 |u| q −1 |υ n|q −1 |υ| q −1|z|dx. 3.21
... 3.11 Also as we know, the operator F < /p>r : W p x ,q x → W∗ < /p>
p x ,q x is compact 3 Then, for. .. < /p>
b2 x |υ|q< /small> −1|z|dx, < /p>
3.6 < /p>
and applying Propositions2. 1,2 . 3, and2.6and considering the conditions p x > p x and < /p>
q x ... q< /small> < /p>
q x < /p>
< /p>
< ∞. < /p>
3.5 < /p> Trang 8
Hence,F