Directory UMM :Journals:Journal_of_mathematics:EJQTDE:
Electronic Journal of Qualitative Theory of Differential Equations
2010, No. 65, 1-13; http://www.math.u-szeged.hu/ejqtde/
Existence of solutions for
p(x)−Laplacian
equations1
R. A. Mashiyeva,2 , B. Cekica and O. M. Buhriib
a
b
Department of Mathematics, Dicle University 21280-Diyarbakir, Turkey.
Department of Differential Equations, Ivan Franko National University of Lviv,
79000-Lviv, Ukraine
[email protected] & [email protected] & ol [email protected]
Abstract
We discuss the problem
(
p(x)−2
q(x)−2
h(x)−2
− div |∇u|
∇u = λ(a (x) |u|
u + b(x) |u|
u), for x ∈ Ω,
u = 0,
for x ∈ ∂Ω.
where Ω is a bounded domain with smooth boundary in RN (N ≥ 2) and p is Lipschitz continuous, q and h are continuous functions on Ω such that 1 < q(x) < p(x) <
h(x) < p∗ (x) and p(x) < N . We show the existence of at least one nontrivial weak
solution. Our approach relies on the variable exponent theory of Lebesgue and Sobolev
spaces combined with adequate variational methods and the Mountain Pass Theorem.
Keywords and Phrases: variable exponent Lebesgue and Sobolev spaces; p (x)-Laplacian;
variational methods; Mountain Pass Theorem.
AMS Subject Classifications (2000): 35J60, 35B30, 35B40
1. Introduction
The study of partial differential equations and variational problems involving p(x)-growth conditions has captured special attention in the last decades. This is a consequence of the fact that
such equations can be used to model phenomena which arise in mathematical physics, for example:
• Electrorheological fluids: see Acebri and Mingione [1], Zhikov [25] and R˚
uˇziˇcka [20], Fan and
Zhang [12], Mih˘
ailescu and R˘
adulescu [16], Chabrowski and Fu [7], H¨ast¨o [14], Diening [8].
• Nonlinear porous medium: see Antontsev and Rodrigues [2], Buhrii and Mashiyev [4], and
Songzhe, Wenjie, Chunling and Hongjun [21].
• Image Processing: Chen, Levine and Rao [6].
A typical model of an elliptic equation with p(x)-growth conditions is
p(x)−2
− div |∇u|
∇u = f (x, u).
(1.1)
p(x)−2
The operator − div |∇u|
∇u is called the p(x)-Laplace operator and it is a natural generalization of the p−Laplace operator, in which p(x) ≡ p > 1 is a constant. The p(x)-Laplacian
1 This
research was supported by DUBAP grant No. 10-FF-15, Dicle University, Turkey.
2 Corresponding
author, E-mail: [email protected]
EJQTDE, 2010 No. 65, p. 1
processes have more complicated nonlinearity, for example, it is nonhomogeneous, so in the discussions some special techniques will be needed.
Problems like (1.1) with Dirichlet boundary condition have been largely considered in the literature in the recent years. We give in what follows a concise but complete image of the actual stage
of research on this topic. We will use the notations such as p1 and p2 where
p1 := ess inf p (x) ≤ p (x) ≤ p2 := ess sup p (x) < ∞.
x∈Ω
x∈Ω
p(x)−2
In the case f (x, u) = λ |u|
u in [13] the authors established the existence of infinitely many
eigenvalues for problem (1.1) by using an argument based on the Ljusternik-Schnirelmann critical
point theory. Denoting by Λ the set of all nonnegative eigenvalues, they showed that Λ is discrete,
sup Λ = ∞ and pointed out inf Λ = 0 for general p(x), and only under some special conditions
inf Λ > 0. In the case f (x, u) = λ |u|q(x)−2 u, there are different papers, for example, in [12] the
same authors proved that any λ > 0 is an eigenvalue of problem (1.1) when p2 < q1 and also when
q2 < p1 . In [18] the authors proved the existence of a continuous family of eigenvalues which lies in
a neighborhood of the origin when q1 < p1 and q(x) has subcritical growth in problem (1.1).
a−2
b−2
p1
In the case f (x, u) = A |u|
u + B |u|
u with 1 < a < p1 < p2 < b < min{N, NN−p
} and
1
A, B > 0, in [17] Mih˘
ailescu show that there exists λ > 0 such that, for any A, B ∈ (0, λ), problem
(1.1) has at least two distinct nontrivial weak solutions.
The aim of this paper is to discuss the existence of a weak solution of the p (x)-Laplacian
equation
(
− div |∇u|p(x)−2 ∇u = λ(a (x) |u|q(x)−2 u + b(x) |u|h(x)−2 u), for x ∈ Ω,
(Pλ )
u = 0,
for x ∈ ∂Ω.
where Ω ⊂ RN (N ≥ 2) is a bounded domain with smooth boundary, λ is a positive real number,
p is Lipschitz continuous on Ω, and q, h ∈ C+ (Ω), a(x), b(x) > 0 for x ∈ Ω such that a ∈ Lβ(x) (Ω),
p∗ (x)
p(x)
p(x)
and b ∈ Lγ(x) (Ω), γ(x) = p∗ (x)−h(x)
. Here p∗ (x) = NN−p(x)
if p(x) < N or
β(x) = p(x)−q(x)
∗
p (x) = ∞ if p(x) ≥ N.
In the present paper, assuming the condition
1 < q1 ≤ q2 < p1 ≤ p2 < h1 ≤ h2 < p∗1 and p2 < N
(1.2)
and using the the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem, we show the existence of at least one
nontrivial weak solution of problem (Pλ ).
2.Preliminaries
We recall in what follows some definitions and basic properties of variable exponent Lebesgue
1,p(x)
(Ω). In that context we refer to [9, 10, 15]
and Sobolev spaces Lp(x) (Ω), W 1,p(x) (Ω) and W0
for the fundamental properties of these spaces.
Set
EJQTDE, 2010 No. 65, p. 2
L∞
+
∞
(Ω) = p; p ∈ L (Ω) , ess inf p(x) > 1 .
x∈Ω
p(.)
For p ∈ L∞
(Ω) to consist of all measurable
+ (Ω) , we define the variable exponent Lebesgue space L
functions u : Ω → R for which the modular
Z
ρp(x) (u) = |u (x)|p(x) dx
Ω
is finite. We define the Luxembourg norm on this space by the formula
o
n
u
≤1 .
kukp(x) = inf δ > 0 : ρp(x)
δ
Equipped with this norm, Lp(.) (Ω) is a separable and reflexive Banach space. Define the variable
exponent Sobolev space W 1,p(x) (Ω) by
W 1,p(x) (Ω) = {u ∈ Lp(x) (Ω) ; |∇u| ∈ Lp(x) (Ω)},
and the norm
kuk1,p(x) = kukp(x) + k∇ukp(x) , ∀u ∈ W 1,p(x) (Ω)
1,p(x)
makes W 1,p(x) (Ω) a separable and reflexive Banach space. The space W0
(Ω) is denoted by the
1,p(x)
∞
1,p(x)
closure of C0 (Ω) in W
(Ω). W0
(Ω) is a separable and reflexive Banach space.
1
1
Proposition 2.1 [10, 15] The conjugate space of Lp(x) (Ω) is Lq(x) (Ω), where p(x)
+ q(x)
= 1.
For any u ∈ Lp(x) (Ω) and υ ∈ Lq(x) (Ω), we have
Z
uυdx ≤ 1 + 1 kuk
p(x) kυkq(x) ≤ 2 kukp(x) kυkq(x) .
p1
q1
Ω
The next proposition illuminates the close relation between the k·kp(x) and the convex modular
ρp(x) :
Proposition 2.2 [9, 10, 15] If u ∈ Lp(x) (Ω) and p2 < ∞ then we have
i) kukp(x)
< 1 (= 1; > 1) ⇔ ρp(x) (u) < 1 (= 1; > 1) ,
p
p
p
p
ii) kukp(x)
1
2
> 1 =⇒ kukp(x)
≤ ρp(x) (u) ≤ kukp(x)
,
iii) kukp(x)
2
1
< 1 =⇒ kukp(x)
≤ ρp(x) (u) ≤ kukp(x)
,
u
=1
= a > 0 ⇐⇒ ρp(x)
a
iv) kukp(x)
Proposition 2.3 [9, 10, 15] If u, un ∈ Lp(x) (Ω) , n = 1, 2, ..., then the following statements are
equivalent
(1) lim kun − ukp(x) = 0;
n→∞
(2) lim ρp(x) (un − u) = 0;
n→∞
(3) un → u in measure in Ω and lim ρp(x) (un ) = ρp(x) (u).
n→∞
EJQTDE, 2010 No. 65, p. 3
r(x)
Lemma 2.4 [5] Assume that r ∈ L∞
∈
+ (Ω) and p ∈ C+ (Ω) := m ∈ C(Ω) : m1 > 1 . If |u|
p(x)
L
(Ω), then we have
n
o
1
2
min kukrr(x)p(x)
, kukrr(x)p(x)
≤
|u|r(x)
Remark 2.5 If r(x) ≡ r, r ∈ R then
r
p(x)
n
o
1
2
≤ max kukrr(x)p(x)
, kukrr(x)p(x)
.
r
k|u| kp(x) = kukrp(x) .
Given two Banach spaces X and Y , the symbol X ֒→ Y means that X is continuously imbedded
in Y and the symbol X ֒→֒→ Y means that there is a compact embedding of X in Y .
Proposition 2.6 [9, 10, 11, 15] Assume that Ω is bounded and smooth.
(i) Let q, h ∈ C+ (Ω). If q(x) ≤ h(x) for all x ∈ Ω, then Lh(x) (Ω) ֒→ Lq(x) (Ω) .
∗
(ii) Let p is Lipschitz continuous and p2 < N, then for h ∈ L∞
+ (Ω) with p(x) ≤ h(x) ≤ p (x)
there is a continuous imbedding W 1,p(x) (Ω) ֒→ Lh(x) (Ω) , and also there is a constant C1 > 0 such
that kukh(x) ≤ C1 kuk1,p(x) .
(iii) Let p, q ∈ C+ (Ω). If p(x) ≤ q(x) < p∗ (x) for all x ∈ Ω, then W 1,p(x) (Ω) ֒→֒→ Lq(x) (Ω) .
(iv) (Poincar´e inequality ) If p ∈ C+ (Ω), then there is a constant C2 > 0 such that
1,p(x)
kukp(x) ≤ C2 k|∇u|kp(x) , ∀ u ∈ W0
(Ω) .
1,p(x)
Consequently, kuk := k|∇u|kp(x) and kuk1,p(x) are equivalent norms on W0
1,p(x)
lows, W0
(Ω). In what fol-
(Ω), with p ∈ C+ (Ω), will be considered as endowed with the norm kuk1,p(x) . We will
1,p(x)
use kuk = k∇ukp(x) for u ∈ W0
(Ω) in the following discussions.
Finally, we introduce Mountain-Pass Theorem which is the main tool of the present paper.
Palais-Smale condition [24] Let E be a Banach space and I ∈ C 1 (E, R). If {un } ⊂ E is a
sequence which satisfies conditions
|Iλ (un )| < M,
Iλ′ (un ) → 0 as n → ∞ in E ∗
where M is a positive constant and E ∗ is the dual space of E, then {un } possesses a convergent
subsequence.
Mountain-Pass Theorem [24] Let E be a Banach space, and let I ∈ C 1 (E, R) satisfy the
Palais-Smale condition. Assume that I (0) = 0, and there exists a positive real number ρ and
u, υ ∈ E such that
(i) kυk > ρ, I (υ) ≤ I (0) .
(ii) α = inf {I (u) : u ∈ E, kuk = ρ} > 0.
Put G = {g ∈ C ([0, 1] , E) : g (0) = 0, g (1) = υ} 6= ∅. Set β = inf sup I (g (t)) .
g∈G t∈[0,1]
EJQTDE, 2010 No. 65, p. 4
Then, β ≥ α and β is a critical value of I.
3. Main Results
1,p(x)
The energy functional corresponding to problem (Pλ ) is defined as Jλ : W0
Z
Jλ (u) =
p(x)
|∇u|
dx − λ
p(x)
Z
a(x) q(x)
|u|
dx − λ
q(x)
Ω
Ω
Z
(Ω) → R,
b(x) h(x)
|u|
dx.
h(x)
(3.1)
Ω
1,p(x)
W0
We say that u ∈
(Ω) is a weak solution for problem (Pλ ) provided
Z
Z
Z
p(x)−2
q(x)−2
h(x)−2
|∇u|
∇u∇υdx = λ a(x) |u|
uυdx + λ b(x) |u|
uυdx
Ω
Ω
1,p(x)
for all υ ∈ W0
Ω
(Ω) .
1,p(x)
(Ω) , R with
Standard arguments imply that Jλ ∈ C 1 W0
hJλ′ (u), υi
=
Z
p(x)−2
|∇u|
∇u∇υdx − λ
Ω
1,p(x)
for all u, υ ∈ W0
Z
q(x)−2
a(x) |u|
uυdx − λ
Ω
Z
h(x)−2
b(x) |u|
uυdx
(3.2)
Ω
(Ω). Thus the weak solution of (Pλ ) are exactly the critical points of Jλ .
The main result of the present paper is the following theorem.
Theorem 3.1 Assume p is Lipschitz continuous, q, h ∈ C+ (Ω) and condition (1.2) is fulfilled.
If
λ∈
0, min
(
h1 ρp2 −h1
q1 ρp2 −q1
,
4C1q1 p2 kakβ(x) 4C1h1 p2 kbkγ(x)
)!
,
then the problem (Pλ ) has at least one nontrivial solution, where ρ ∈ (0, 1) .
To obtain the proof of Theorem 3.1, we use Mountain-Pass theorem. Therefore, we must show
Jλ satisfies Palais-Smale condition in the first place.
1,p(x)
Lemma 3.2 Let λ satisfies the condition of Theorem 3.1. If {un } ⊂ W0
which satisfies conditions
|Jλ (un )| < M,
Jλ′ (un ) → 0 as n → ∞ in
∗
1,p(x)
W0
(Ω)
(Ω) is a sequence
(3.3)
(3.4)
where M is a positive constant, then {un } possesses a convergent subsequence.
1,p(x)
(Ω). Assume the contrary. Then, passing
Proof: First, we show that {un } is bounded in W0
to a subsequence if necessary, we may assume that kun k → ∞ as n → ∞. Thus, we may consider
EJQTDE, 2010 No. 65, p. 5
that kun k > 1 for any integer n. By (3.4) we deduce that there exists N1 > 0 such that for any
n > N1 , we have
kJλ′ (un )k ≤ 1.
On the other hand, for any n > N1 fixed, the application
1,p(x)
W0
(Ω) ∋ υ → hJλ′ (un ), υi
is linear and continuous. The above information implies
1,p(x)
|hJλ′ (un ), υi| ≤ kJλ′ (un )kW −1,p′ (x) (Ω) kυk ≤ kυk , ∀υ ∈ W0
0
(Ω) , n > N1 .
Setting υ = un we have
Z
Z
Z
p(x)
q(x)
h(x)
− kun k ≤ |∇un |
dx − λ a(x) |un |
dx − λ b(x) |un |
dx ≤ kun k
Ω
Ω
Ω
for any n > N1 .
Using the assumption kun k > 1, relations (3.3) , (3.4), Proposition 2.1, Lemma 2.4 and Proposition 2.6 (ii) we have
M
1
hJ ′ (un ), un i
h1 λ
Z
Z
Z
|∇un |p(x)
a(x)
b(x)
q(x)
=
dx − λ
|un |
dx − λ
|un |h(x) dx
p(x)
q(x)
h(x)
Ω
Ω
Ω
Z
Z
Z
λ
λ
1
p(x)
q(x)
h(x)
|∇un |
dx +
a(x) |un |
dx +
b(x) |un |
dx
−
h1
h1
h1
Ω
Ω
Ω
Z
Z
Z
λ
λ
1
p(x)
q(x)
h(x)
|∇un |
dx −
a(x) |un |
dx −
b(x) |un |
dx
≥
p2
q1
h1
Ω
Ω
Ω
Z
Z
Z
1
λ
λ
p(x)
q(x)
h(x)
−
|∇un |
dx +
a(x) |un |
dx +
b(x) |un |
dx
h1
h1
h1
Ω
Ω
Ω
1
1
1
1
p1
q
≥
−
−
kun k − λ
C3 kakβ(x) kun k 2 ,
p2
h1
h1
q1
> Jλ (un ) −
(3.5)
where C3 > 0 is a constant independent of un and x, for n large enough. Dividing (3.5) by kun kp1
and passing to the limit as n → ∞ we obtain p12 − h11 < 0.
1,p(x)
Since q1 ≤ q2 < p1 ≤ p2 < h1 , this is a contradiction. It follows {un } is bounded in W0
(Ω) .)
1,p(x)
Next, we show the strong convergence of {un } in W0
(Ω). Since {un } is bounded, up to a
1,p(x)
subsequence (which we still denote by {un }), we may assume that there exists u ∈ W0
(Ω) such
that
1,p(x)
un ⇀ u weakly in W0
(Ω) as n → ∞.
EJQTDE, 2010 No. 65, p. 6
By Proposition 2.6 (iii) we obtain
un → u strongly in Lp(x) (Ω) as n → ∞.
(3.6)
Furthermore, from [3, 23] we have
∗
un → u strongly in Lp
(x)
(K) as n → ∞,
(3.7)
where K is compact subset of Ω.
The above information and relation (3.4) imply
hJλ′ (un ) − Jλ′ (u), un − ui → 0 as n → ∞.
On the other hand, we have
Z
p(x)−2
p(x)−2
|∇un |
∇un − |∇u|
∇u (∇un − ∇u) dx
Ω
=
hJλ′ (un )
−
Jλ′ (u), un
− ui − λ
Z
Ω
−λ
Z
h(x)−2
b(x) |un |
Ω
a(x) |un |q(x)−2 un − |u|q(x)−2 u (un − u) dx
h(x)−2
un − |u|
u (un − u) dx.
Propositions 2.1, 2.3 and Lemma 2.4 we have
Z
q(x)−2
q(x)−2
λ a(x) |un |
un − |u|
u1 (un − u) dx
Ω
Z
Z
q(x)−2
q(x)−2
≤ λ a(x) |un |
un (un − u) dx + λ a(x) |u1 |
u (un − u) dx
Ω
Ω
q(x)−1
q(x)−1
≤ C4 kakβ(x)
|un |
p(x) kun − ukp(x)
p(x) kun − ukp(x) + C5 kakβ(x) |u|
q(x)−1
≤
q2 −1
C4 kakβ(x) kun kp(x)
q(x)−1
kun − ukp(x) +
2 −1
C5 kakβ(x) kukqp(x)
kun − ukp(x) ,
1
1
+ q(x)−1
where C4 , C5 > 0 and β(x)
p(x) + p(x) = 1.
Similarly, Propositions 2.1, 2.3 and Lemma 2.4 we have
Z
h(x)−2
h(x)−2
λ b(x) |un |
un − |u|
u (un − u) dx
Ω
h(x)−1
h(x)−1
≤ C6 kbkγ(x)
|un |
p∗ (x) kun − ukp∗ (x) + C7 kbkγ(x)
|u|
h(x)−1
≤
h −1
p∗ (x)
h(x)−1
kun − ukp∗ (x)
h −1
C6 kbkγ(x) ||un ||p∗2(x) kun − ukp∗ (x) + C7 kbkγ(x) kukp∗2(x) kun − ukp∗ (x) ,
where C6 , C7 > 0 and
1
γ(x)
+
h(x)−1
p∗ (x)
+
1
p∗ (x)
= 1. By (3.6) and (3.7) we have
EJQTDE, 2010 No. 65, p. 7
kun − ukp(x) → 0 as n → ∞
and
kun − ukp∗ (x) → 0 as n → ∞.
Therefore, from above inequalities we deduce that
Z
h(x)−2
h(x)−2
lim
b(x) |un |
un − |u|
u (un − u) = 0,
n→∞
(3.8)
Ω
and
lim
n→∞
Z
Ω
q(x)−2
q(x)−2
a(x) |un |
un − |u|
u (un − u) dx = 0,
respectively. By (3.8) and (3.9) we obtain
Z
p(x)−2
p(x)−2
lim
|∇un |
∇un − |∇u|
∇u (∇un − ∇u) = 0.
n→∞
(3.9)
(3.10)
Ω
This result and the following inequality [23, Lemma 2.2]
r−2
r−2
|ξ|
ξ − |η|
η (ξ − η) ≥ 2−r |ξ − η| , ∀r ≥ 2; ξ, η ∈ RN .
(3.11)
Z
(3.12)
yield
lim
n→∞
p(x)
|∇un − ∇u|
dx = 0.
Ω
This fact and Proposition 2.3 imply k∇un − ∇ukp(x) → 0 as n → ∞. Relation (3.12) and fact that
1,p(x)
un ⇀ u (weakly) in W0
(Ω) enable us to apply [12] in order to obtain that un ⇀ u (strongly)
1,p(x)
in W0
(Ω). Thus, Lemma 3.2 is proved.
Now, we show that the Mountain-Pass theorem can be applied in this case.
Lemma 3.3 Assume p, q, h ∈ C+ (Ω) and condition (1.2) is fulfilled. The following assertions
hold.
(i) There exist λ > 0, α > 0 and ρ ∈ (0, 1) such that
1,p(x)
Jλ (u) ≥ α, ∀u ∈ W0
(ii) There exists ω ∈
1,p(x)
W0
(Ω)
(Ω) with kuk = ρ.
such that
lim Jλ (tω) = −∞.
t→∞
1,p(x)
(iii) There exists ϕ ∈ W0
(3.13)
(3.14)
(Ω) such that ϕ ≥ 0, ϕ 6= 0 and
Jλ (tϕ) < 0,
(3.15)
EJQTDE, 2010 No. 65, p. 8
for t > 0 small enough.
Proof. (i) Using Propositions 2.1, 2.2, 2.6 (ii) and Lemma 2.4 we deduce that for any u ∈
(Ω) with ρ ∈ (0, 1) we have
Z
Z
Z
λ
λ
1
p(x)
q(x)
h(x)
|∇u|
dx −
a(x) |u|
dx −
b(x) |u|
dx
Jλ (u) ≥
p2
q1
h1
1,p(x)
W0
Ω
≥
≥
≥
Ω
Ω
λ
1
λ
q(x)
h(x)
p
kuk 2 −
kakβ(x)
|u|
kbkγ(x)
|u|
p(x) −
p∗ (x)
p2
q1
h1
q(x)
h(x)
λ
λ
1
p2
q1
h1
kuk −
kakβ(x) kukp(x) −
kbkγ(x) kukp∗ (x)
p2
q1
h1
λ
λ
1
kukp2 − C1q1 kakβ(x) kukq1 − C1h1
kbkγ(x) kukh1 .
p2
q1
h1
Taking
λ = min
we obtain
Jλ (u) ≥
(
h1 ρp2 −h1
q1 ρp2 −q1
,
q1
4C1 p2 kakβ(x) 4C1h1 p2 kbkγ(x)
)
ρp2
1,p(x)
, ∀u ∈ W0
(Ω) with kuk = ρ.
2p2
Thus Lemma 3.3 (i) is proved.
(ii) Let ω ∈ C0∞ (Ω) , ω ≥ 0, ω 6= 0 and t > 1.We have
Jλ (tω)
=
≤
Z q(x)
Z h(x)
t
t
tp(x)
p(x)
q(x)
h(x)
|∇ω|
dx − λ
a(x) |ω|
dx − λ
b(x) |ω|
dx
p(x)
q(x)
h(x)
Ω
Ω
Ω
Z
Z
Z
tp2
tq2
th 2
p(x)
q(x)
h(x)
|∇ω|
dx − λ
a(x) |ω|
dx − λ
b(x) |ω|
dx.
p1
q1
h1
Z
Ω
Ω
Ω
Since q2 , p2 < h2 we have Jλ (tω) → −∞. Thus Lemma 3.3 (ii) is proved.
(iii) Let ϕ ∈ C0∞ (Ω) , ϕ ≥ 0, ϕ 6= 0 and t ∈ (0, 1). We have
Z
Z
Z
tp1
tq1
th 1
p(x)
q(x)
h(x)
Jλ (tϕ) ≤
|∇ϕ|
dx − λ
a(x) |ϕ|
dx − λ
b(x) |ϕ|
dx.
p1
q1
h1
Ω
Ω
Ω
Since
λtq1
q1
<
λtq1
q1
Z
a(x) |ϕ|q(x) dx +
Ω
λth1
h1
Z
b(x) |ϕ|h(x) dx
Ω
Z
Z
a(x) |ϕ|q(x) dx + b(x) |ϕ|h(x) dx ,
Ω
Ω
EJQTDE, 2010 No. 65, p. 9
we have
tp1
Jλ (tϕ) ≤
p1
for t < δ
1
p1 −q1
with
Z
Ω
Z
Z
q1
λt
p(x)
a(x) |ϕ|q(x) dx + b(x) |ϕ|h(x) dx < 0,
|∇ϕ|
dx −
q1
Ω
Ω
Z
Z
q(x)
h(x)
λp1 a(x) |ϕ|
dx + b(x) |ϕ|
dx
Ω
Ω
Z
.
0 < δ < min 1,
p(x)
q
|∇ϕ|
dx
1
Ω
Lemma 3.3 (iii) is proved.
Proof of Theorem 3.1 We set
n
o
1,p(x)
G = g ∈ C([0, 1] , W0
(Ω)) : g (0) = 0, g (1) = υ ,
1,p(x)
where υ ∈ W0
(Ω) is determined by Lemma 3.3 (ii) and (iii), and
β := inf sup Jλ (g (t)) .
g∈G t∈[0,1]
According to Lemma 3.3 (ii) and (iii), we know that kυk > ρ, so every path g ∈ G intersects
the sphere kvk = ρ. Then Lemma 3.3 (i) implies
α ≤ inf Jλ (u) ≤ β,
kuk=ρ
with the constant α > 0 in Lemma 3.3 (i), thus β > 0. By the Mountain-Pass theorem Jλ admits
a critical value β ≥
α.
1,p(x)
(Ω) , R , from Lemma 3.2 we conclude
Since Jλ ∈ C 1 W0
Jλ′ (un ) → Jλ′ (u)
(3.16)
as n → ∞.
Relations (3.3), (3, 4) and (3.16) show that Jλ′ (u) = 0 and thus u is a weak solutions for problem
(Pλ ). Moreover, by relations (3.3), (3, 4) it follows that Jλ (u) > 0 and thus, u is a nontrivial weak
solutions for problem (Pλ ). The proof is completed.
Remark 3.4 If (u, λ) is a solution of (Pλ ) and u 6= 0, as usual, we call λ and u an eigenvalue
eigenfunction corresponding to λ of (Pλ ), respectively. If (u, λ) is a solution of (Pλ ) and u 6= 0,
then
Z
p(x)
|∇u|
dx
λ = λ(u) = Z
Ω
q(x)
(a(x) |u|
h(x)
+ b(x) |u|
)dx
Ω
EJQTDE, 2010 No. 65, p. 10
and hence λ > 0.
Theorem 3.1 ensures that problem (Pλ ) has a continuous family of positive eigenvalues that lie
in a neighborhood of the origin. Furthermore, we obtain
Z
p(x)
|∇u|
dx
inf
Ω
1,p(x)
u∈W0
\{0}
Z
q(x)
(a(x) |u|
h(x)
+ b(x) |u|
= 0.
)dx
Ω
Remark 3.5 Furthermore, we can consider the equation
(
− div |∇u|p(x)−2 ∇u = λ(a (x) |u|q(x)−2 u − b(x) |u|h(x)−2 u), for x ∈ Ω,
u = 0,
for x ∈ ∂Ω.
/
(Pλ )
where p, q, h, a, b and λ are the same in problem (Pλ ).
/
1,p(x)
The energy functional corresponding to problem (Pλ ) is defined as Iλ : W0
(Ω) → R,
Iλ (u) =
|∇u|p(x)
dx − λ
p(x)
Z
Z
a(x) q(x)
|u|
dx + λ
q(x)
Ω
Ω
We infer that for any x ∈ Ω and u ∈
≤
≤
≤
1,p(x)
W0
Z
b(x)
|u|h(x) dx.
h(x)
Ω
(Ω)
λb(x)
λa(x)
q(x)
h(x)
|u(x)|
−
|u(x)|
q(x)
h(x)
λa2
λb1
|u(x)|q(x) −
|u(x)|h(x)
q1
h2
q(x)
λa2 a2 h2 h(x)−q(x)
q1
b1 q1
"
q2
q1 #
λa2
a2 h2 h2 −q1
a2 h2 h1 −q2
:= A
+
q1
b1 q1
b1 q1
(3.17)
where A is a positive constant independent of u and x, by using the following elementary inequality
[19, Lemma 4]
atk − bts ≤ a
k
a s−k
b
,
for all t ≥ 0
where a, b > 0 and 0 < k < s.
Integrating (3.17) over Ω, we get
Z
Z
a(x)
b(x)
q(x)
h(x)
λ
|u(x)|
dx − λ
|u(x)|
dx ≤ D
q(x)
h(x)
Ω
Ω
where D is a positive constant independent of u. Thus,
EJQTDE, 2010 No. 65, p. 11
Jλ (u) ≥
≥
1
p2
Z
p(x)
|∇u|
Ω
dx − λ
Z
Ω
a(x) q(x)
|u|
dx + λ
q(x)
Z
b(x) h(x)
|u|
dx
h(x)
Ω
1
p
kuk 1 − D,
p2
1,p(x)
for all u ∈ W0
(Ω) with kuk > 1. We infer that Jλ (u) → ∞ as kuk → ∞. Therefore the energy
1,p(x)
functional Iλ is coercive on W0
(Ω). Moreover, a similar argument as the one used in the proof
1,p(x)
(Ω). These facts
of [16, Lemma 3.4] shows that Iλ is also weakly lower semi-continuous in W0
1,p(x)
enable us to apply [22, Theorem 1.2] in order to find that there exits uλ ∈ W0
(Ω) a global
/
minimizer of Iλ and thus, a weak solution of problem (Pλ ).
Acknowledgments
The authors would like to thank the referees and Professor Paul Eloe for their many helpful
suggestions and corrections, which improve the presentation of this paper.
References
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akosnik, Sobolev embeddings with variable exponent, Studia Math. 143
(2000), 267-293.
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[10] X. L. Fan and D. Zhao, On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263
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[15] O. Kov˘aˇcik and J. R˘
akosnik, On spaces Lp(x) and W k,p(x) , Czechoslovak Math. J. 41 (116)
(1991), 592–618.
[16] M. Mih˘ailescu and V. R˘
adulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proceedings of the Royal Society A: Mathematical,
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(Received January 27, 2010)
EJQTDE, 2010 No. 65, p. 13
2010, No. 65, 1-13; http://www.math.u-szeged.hu/ejqtde/
Existence of solutions for
p(x)−Laplacian
equations1
R. A. Mashiyeva,2 , B. Cekica and O. M. Buhriib
a
b
Department of Mathematics, Dicle University 21280-Diyarbakir, Turkey.
Department of Differential Equations, Ivan Franko National University of Lviv,
79000-Lviv, Ukraine
[email protected] & [email protected] & ol [email protected]
Abstract
We discuss the problem
(
p(x)−2
q(x)−2
h(x)−2
− div |∇u|
∇u = λ(a (x) |u|
u + b(x) |u|
u), for x ∈ Ω,
u = 0,
for x ∈ ∂Ω.
where Ω is a bounded domain with smooth boundary in RN (N ≥ 2) and p is Lipschitz continuous, q and h are continuous functions on Ω such that 1 < q(x) < p(x) <
h(x) < p∗ (x) and p(x) < N . We show the existence of at least one nontrivial weak
solution. Our approach relies on the variable exponent theory of Lebesgue and Sobolev
spaces combined with adequate variational methods and the Mountain Pass Theorem.
Keywords and Phrases: variable exponent Lebesgue and Sobolev spaces; p (x)-Laplacian;
variational methods; Mountain Pass Theorem.
AMS Subject Classifications (2000): 35J60, 35B30, 35B40
1. Introduction
The study of partial differential equations and variational problems involving p(x)-growth conditions has captured special attention in the last decades. This is a consequence of the fact that
such equations can be used to model phenomena which arise in mathematical physics, for example:
• Electrorheological fluids: see Acebri and Mingione [1], Zhikov [25] and R˚
uˇziˇcka [20], Fan and
Zhang [12], Mih˘
ailescu and R˘
adulescu [16], Chabrowski and Fu [7], H¨ast¨o [14], Diening [8].
• Nonlinear porous medium: see Antontsev and Rodrigues [2], Buhrii and Mashiyev [4], and
Songzhe, Wenjie, Chunling and Hongjun [21].
• Image Processing: Chen, Levine and Rao [6].
A typical model of an elliptic equation with p(x)-growth conditions is
p(x)−2
− div |∇u|
∇u = f (x, u).
(1.1)
p(x)−2
The operator − div |∇u|
∇u is called the p(x)-Laplace operator and it is a natural generalization of the p−Laplace operator, in which p(x) ≡ p > 1 is a constant. The p(x)-Laplacian
1 This
research was supported by DUBAP grant No. 10-FF-15, Dicle University, Turkey.
2 Corresponding
author, E-mail: [email protected]
EJQTDE, 2010 No. 65, p. 1
processes have more complicated nonlinearity, for example, it is nonhomogeneous, so in the discussions some special techniques will be needed.
Problems like (1.1) with Dirichlet boundary condition have been largely considered in the literature in the recent years. We give in what follows a concise but complete image of the actual stage
of research on this topic. We will use the notations such as p1 and p2 where
p1 := ess inf p (x) ≤ p (x) ≤ p2 := ess sup p (x) < ∞.
x∈Ω
x∈Ω
p(x)−2
In the case f (x, u) = λ |u|
u in [13] the authors established the existence of infinitely many
eigenvalues for problem (1.1) by using an argument based on the Ljusternik-Schnirelmann critical
point theory. Denoting by Λ the set of all nonnegative eigenvalues, they showed that Λ is discrete,
sup Λ = ∞ and pointed out inf Λ = 0 for general p(x), and only under some special conditions
inf Λ > 0. In the case f (x, u) = λ |u|q(x)−2 u, there are different papers, for example, in [12] the
same authors proved that any λ > 0 is an eigenvalue of problem (1.1) when p2 < q1 and also when
q2 < p1 . In [18] the authors proved the existence of a continuous family of eigenvalues which lies in
a neighborhood of the origin when q1 < p1 and q(x) has subcritical growth in problem (1.1).
a−2
b−2
p1
In the case f (x, u) = A |u|
u + B |u|
u with 1 < a < p1 < p2 < b < min{N, NN−p
} and
1
A, B > 0, in [17] Mih˘
ailescu show that there exists λ > 0 such that, for any A, B ∈ (0, λ), problem
(1.1) has at least two distinct nontrivial weak solutions.
The aim of this paper is to discuss the existence of a weak solution of the p (x)-Laplacian
equation
(
− div |∇u|p(x)−2 ∇u = λ(a (x) |u|q(x)−2 u + b(x) |u|h(x)−2 u), for x ∈ Ω,
(Pλ )
u = 0,
for x ∈ ∂Ω.
where Ω ⊂ RN (N ≥ 2) is a bounded domain with smooth boundary, λ is a positive real number,
p is Lipschitz continuous on Ω, and q, h ∈ C+ (Ω), a(x), b(x) > 0 for x ∈ Ω such that a ∈ Lβ(x) (Ω),
p∗ (x)
p(x)
p(x)
and b ∈ Lγ(x) (Ω), γ(x) = p∗ (x)−h(x)
. Here p∗ (x) = NN−p(x)
if p(x) < N or
β(x) = p(x)−q(x)
∗
p (x) = ∞ if p(x) ≥ N.
In the present paper, assuming the condition
1 < q1 ≤ q2 < p1 ≤ p2 < h1 ≤ h2 < p∗1 and p2 < N
(1.2)
and using the the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem, we show the existence of at least one
nontrivial weak solution of problem (Pλ ).
2.Preliminaries
We recall in what follows some definitions and basic properties of variable exponent Lebesgue
1,p(x)
(Ω). In that context we refer to [9, 10, 15]
and Sobolev spaces Lp(x) (Ω), W 1,p(x) (Ω) and W0
for the fundamental properties of these spaces.
Set
EJQTDE, 2010 No. 65, p. 2
L∞
+
∞
(Ω) = p; p ∈ L (Ω) , ess inf p(x) > 1 .
x∈Ω
p(.)
For p ∈ L∞
(Ω) to consist of all measurable
+ (Ω) , we define the variable exponent Lebesgue space L
functions u : Ω → R for which the modular
Z
ρp(x) (u) = |u (x)|p(x) dx
Ω
is finite. We define the Luxembourg norm on this space by the formula
o
n
u
≤1 .
kukp(x) = inf δ > 0 : ρp(x)
δ
Equipped with this norm, Lp(.) (Ω) is a separable and reflexive Banach space. Define the variable
exponent Sobolev space W 1,p(x) (Ω) by
W 1,p(x) (Ω) = {u ∈ Lp(x) (Ω) ; |∇u| ∈ Lp(x) (Ω)},
and the norm
kuk1,p(x) = kukp(x) + k∇ukp(x) , ∀u ∈ W 1,p(x) (Ω)
1,p(x)
makes W 1,p(x) (Ω) a separable and reflexive Banach space. The space W0
(Ω) is denoted by the
1,p(x)
∞
1,p(x)
closure of C0 (Ω) in W
(Ω). W0
(Ω) is a separable and reflexive Banach space.
1
1
Proposition 2.1 [10, 15] The conjugate space of Lp(x) (Ω) is Lq(x) (Ω), where p(x)
+ q(x)
= 1.
For any u ∈ Lp(x) (Ω) and υ ∈ Lq(x) (Ω), we have
Z
uυdx ≤ 1 + 1 kuk
p(x) kυkq(x) ≤ 2 kukp(x) kυkq(x) .
p1
q1
Ω
The next proposition illuminates the close relation between the k·kp(x) and the convex modular
ρp(x) :
Proposition 2.2 [9, 10, 15] If u ∈ Lp(x) (Ω) and p2 < ∞ then we have
i) kukp(x)
< 1 (= 1; > 1) ⇔ ρp(x) (u) < 1 (= 1; > 1) ,
p
p
p
p
ii) kukp(x)
1
2
> 1 =⇒ kukp(x)
≤ ρp(x) (u) ≤ kukp(x)
,
iii) kukp(x)
2
1
< 1 =⇒ kukp(x)
≤ ρp(x) (u) ≤ kukp(x)
,
u
=1
= a > 0 ⇐⇒ ρp(x)
a
iv) kukp(x)
Proposition 2.3 [9, 10, 15] If u, un ∈ Lp(x) (Ω) , n = 1, 2, ..., then the following statements are
equivalent
(1) lim kun − ukp(x) = 0;
n→∞
(2) lim ρp(x) (un − u) = 0;
n→∞
(3) un → u in measure in Ω and lim ρp(x) (un ) = ρp(x) (u).
n→∞
EJQTDE, 2010 No. 65, p. 3
r(x)
Lemma 2.4 [5] Assume that r ∈ L∞
∈
+ (Ω) and p ∈ C+ (Ω) := m ∈ C(Ω) : m1 > 1 . If |u|
p(x)
L
(Ω), then we have
n
o
1
2
min kukrr(x)p(x)
, kukrr(x)p(x)
≤
|u|r(x)
Remark 2.5 If r(x) ≡ r, r ∈ R then
r
p(x)
n
o
1
2
≤ max kukrr(x)p(x)
, kukrr(x)p(x)
.
r
k|u| kp(x) = kukrp(x) .
Given two Banach spaces X and Y , the symbol X ֒→ Y means that X is continuously imbedded
in Y and the symbol X ֒→֒→ Y means that there is a compact embedding of X in Y .
Proposition 2.6 [9, 10, 11, 15] Assume that Ω is bounded and smooth.
(i) Let q, h ∈ C+ (Ω). If q(x) ≤ h(x) for all x ∈ Ω, then Lh(x) (Ω) ֒→ Lq(x) (Ω) .
∗
(ii) Let p is Lipschitz continuous and p2 < N, then for h ∈ L∞
+ (Ω) with p(x) ≤ h(x) ≤ p (x)
there is a continuous imbedding W 1,p(x) (Ω) ֒→ Lh(x) (Ω) , and also there is a constant C1 > 0 such
that kukh(x) ≤ C1 kuk1,p(x) .
(iii) Let p, q ∈ C+ (Ω). If p(x) ≤ q(x) < p∗ (x) for all x ∈ Ω, then W 1,p(x) (Ω) ֒→֒→ Lq(x) (Ω) .
(iv) (Poincar´e inequality ) If p ∈ C+ (Ω), then there is a constant C2 > 0 such that
1,p(x)
kukp(x) ≤ C2 k|∇u|kp(x) , ∀ u ∈ W0
(Ω) .
1,p(x)
Consequently, kuk := k|∇u|kp(x) and kuk1,p(x) are equivalent norms on W0
1,p(x)
lows, W0
(Ω). In what fol-
(Ω), with p ∈ C+ (Ω), will be considered as endowed with the norm kuk1,p(x) . We will
1,p(x)
use kuk = k∇ukp(x) for u ∈ W0
(Ω) in the following discussions.
Finally, we introduce Mountain-Pass Theorem which is the main tool of the present paper.
Palais-Smale condition [24] Let E be a Banach space and I ∈ C 1 (E, R). If {un } ⊂ E is a
sequence which satisfies conditions
|Iλ (un )| < M,
Iλ′ (un ) → 0 as n → ∞ in E ∗
where M is a positive constant and E ∗ is the dual space of E, then {un } possesses a convergent
subsequence.
Mountain-Pass Theorem [24] Let E be a Banach space, and let I ∈ C 1 (E, R) satisfy the
Palais-Smale condition. Assume that I (0) = 0, and there exists a positive real number ρ and
u, υ ∈ E such that
(i) kυk > ρ, I (υ) ≤ I (0) .
(ii) α = inf {I (u) : u ∈ E, kuk = ρ} > 0.
Put G = {g ∈ C ([0, 1] , E) : g (0) = 0, g (1) = υ} 6= ∅. Set β = inf sup I (g (t)) .
g∈G t∈[0,1]
EJQTDE, 2010 No. 65, p. 4
Then, β ≥ α and β is a critical value of I.
3. Main Results
1,p(x)
The energy functional corresponding to problem (Pλ ) is defined as Jλ : W0
Z
Jλ (u) =
p(x)
|∇u|
dx − λ
p(x)
Z
a(x) q(x)
|u|
dx − λ
q(x)
Ω
Ω
Z
(Ω) → R,
b(x) h(x)
|u|
dx.
h(x)
(3.1)
Ω
1,p(x)
W0
We say that u ∈
(Ω) is a weak solution for problem (Pλ ) provided
Z
Z
Z
p(x)−2
q(x)−2
h(x)−2
|∇u|
∇u∇υdx = λ a(x) |u|
uυdx + λ b(x) |u|
uυdx
Ω
Ω
1,p(x)
for all υ ∈ W0
Ω
(Ω) .
1,p(x)
(Ω) , R with
Standard arguments imply that Jλ ∈ C 1 W0
hJλ′ (u), υi
=
Z
p(x)−2
|∇u|
∇u∇υdx − λ
Ω
1,p(x)
for all u, υ ∈ W0
Z
q(x)−2
a(x) |u|
uυdx − λ
Ω
Z
h(x)−2
b(x) |u|
uυdx
(3.2)
Ω
(Ω). Thus the weak solution of (Pλ ) are exactly the critical points of Jλ .
The main result of the present paper is the following theorem.
Theorem 3.1 Assume p is Lipschitz continuous, q, h ∈ C+ (Ω) and condition (1.2) is fulfilled.
If
λ∈
0, min
(
h1 ρp2 −h1
q1 ρp2 −q1
,
4C1q1 p2 kakβ(x) 4C1h1 p2 kbkγ(x)
)!
,
then the problem (Pλ ) has at least one nontrivial solution, where ρ ∈ (0, 1) .
To obtain the proof of Theorem 3.1, we use Mountain-Pass theorem. Therefore, we must show
Jλ satisfies Palais-Smale condition in the first place.
1,p(x)
Lemma 3.2 Let λ satisfies the condition of Theorem 3.1. If {un } ⊂ W0
which satisfies conditions
|Jλ (un )| < M,
Jλ′ (un ) → 0 as n → ∞ in
∗
1,p(x)
W0
(Ω)
(Ω) is a sequence
(3.3)
(3.4)
where M is a positive constant, then {un } possesses a convergent subsequence.
1,p(x)
(Ω). Assume the contrary. Then, passing
Proof: First, we show that {un } is bounded in W0
to a subsequence if necessary, we may assume that kun k → ∞ as n → ∞. Thus, we may consider
EJQTDE, 2010 No. 65, p. 5
that kun k > 1 for any integer n. By (3.4) we deduce that there exists N1 > 0 such that for any
n > N1 , we have
kJλ′ (un )k ≤ 1.
On the other hand, for any n > N1 fixed, the application
1,p(x)
W0
(Ω) ∋ υ → hJλ′ (un ), υi
is linear and continuous. The above information implies
1,p(x)
|hJλ′ (un ), υi| ≤ kJλ′ (un )kW −1,p′ (x) (Ω) kυk ≤ kυk , ∀υ ∈ W0
0
(Ω) , n > N1 .
Setting υ = un we have
Z
Z
Z
p(x)
q(x)
h(x)
− kun k ≤ |∇un |
dx − λ a(x) |un |
dx − λ b(x) |un |
dx ≤ kun k
Ω
Ω
Ω
for any n > N1 .
Using the assumption kun k > 1, relations (3.3) , (3.4), Proposition 2.1, Lemma 2.4 and Proposition 2.6 (ii) we have
M
1
hJ ′ (un ), un i
h1 λ
Z
Z
Z
|∇un |p(x)
a(x)
b(x)
q(x)
=
dx − λ
|un |
dx − λ
|un |h(x) dx
p(x)
q(x)
h(x)
Ω
Ω
Ω
Z
Z
Z
λ
λ
1
p(x)
q(x)
h(x)
|∇un |
dx +
a(x) |un |
dx +
b(x) |un |
dx
−
h1
h1
h1
Ω
Ω
Ω
Z
Z
Z
λ
λ
1
p(x)
q(x)
h(x)
|∇un |
dx −
a(x) |un |
dx −
b(x) |un |
dx
≥
p2
q1
h1
Ω
Ω
Ω
Z
Z
Z
1
λ
λ
p(x)
q(x)
h(x)
−
|∇un |
dx +
a(x) |un |
dx +
b(x) |un |
dx
h1
h1
h1
Ω
Ω
Ω
1
1
1
1
p1
q
≥
−
−
kun k − λ
C3 kakβ(x) kun k 2 ,
p2
h1
h1
q1
> Jλ (un ) −
(3.5)
where C3 > 0 is a constant independent of un and x, for n large enough. Dividing (3.5) by kun kp1
and passing to the limit as n → ∞ we obtain p12 − h11 < 0.
1,p(x)
Since q1 ≤ q2 < p1 ≤ p2 < h1 , this is a contradiction. It follows {un } is bounded in W0
(Ω) .)
1,p(x)
Next, we show the strong convergence of {un } in W0
(Ω). Since {un } is bounded, up to a
1,p(x)
subsequence (which we still denote by {un }), we may assume that there exists u ∈ W0
(Ω) such
that
1,p(x)
un ⇀ u weakly in W0
(Ω) as n → ∞.
EJQTDE, 2010 No. 65, p. 6
By Proposition 2.6 (iii) we obtain
un → u strongly in Lp(x) (Ω) as n → ∞.
(3.6)
Furthermore, from [3, 23] we have
∗
un → u strongly in Lp
(x)
(K) as n → ∞,
(3.7)
where K is compact subset of Ω.
The above information and relation (3.4) imply
hJλ′ (un ) − Jλ′ (u), un − ui → 0 as n → ∞.
On the other hand, we have
Z
p(x)−2
p(x)−2
|∇un |
∇un − |∇u|
∇u (∇un − ∇u) dx
Ω
=
hJλ′ (un )
−
Jλ′ (u), un
− ui − λ
Z
Ω
−λ
Z
h(x)−2
b(x) |un |
Ω
a(x) |un |q(x)−2 un − |u|q(x)−2 u (un − u) dx
h(x)−2
un − |u|
u (un − u) dx.
Propositions 2.1, 2.3 and Lemma 2.4 we have
Z
q(x)−2
q(x)−2
λ a(x) |un |
un − |u|
u1 (un − u) dx
Ω
Z
Z
q(x)−2
q(x)−2
≤ λ a(x) |un |
un (un − u) dx + λ a(x) |u1 |
u (un − u) dx
Ω
Ω
q(x)−1
q(x)−1
≤ C4 kakβ(x)
|un |
p(x) kun − ukp(x)
p(x) kun − ukp(x) + C5 kakβ(x) |u|
q(x)−1
≤
q2 −1
C4 kakβ(x) kun kp(x)
q(x)−1
kun − ukp(x) +
2 −1
C5 kakβ(x) kukqp(x)
kun − ukp(x) ,
1
1
+ q(x)−1
where C4 , C5 > 0 and β(x)
p(x) + p(x) = 1.
Similarly, Propositions 2.1, 2.3 and Lemma 2.4 we have
Z
h(x)−2
h(x)−2
λ b(x) |un |
un − |u|
u (un − u) dx
Ω
h(x)−1
h(x)−1
≤ C6 kbkγ(x)
|un |
p∗ (x) kun − ukp∗ (x) + C7 kbkγ(x)
|u|
h(x)−1
≤
h −1
p∗ (x)
h(x)−1
kun − ukp∗ (x)
h −1
C6 kbkγ(x) ||un ||p∗2(x) kun − ukp∗ (x) + C7 kbkγ(x) kukp∗2(x) kun − ukp∗ (x) ,
where C6 , C7 > 0 and
1
γ(x)
+
h(x)−1
p∗ (x)
+
1
p∗ (x)
= 1. By (3.6) and (3.7) we have
EJQTDE, 2010 No. 65, p. 7
kun − ukp(x) → 0 as n → ∞
and
kun − ukp∗ (x) → 0 as n → ∞.
Therefore, from above inequalities we deduce that
Z
h(x)−2
h(x)−2
lim
b(x) |un |
un − |u|
u (un − u) = 0,
n→∞
(3.8)
Ω
and
lim
n→∞
Z
Ω
q(x)−2
q(x)−2
a(x) |un |
un − |u|
u (un − u) dx = 0,
respectively. By (3.8) and (3.9) we obtain
Z
p(x)−2
p(x)−2
lim
|∇un |
∇un − |∇u|
∇u (∇un − ∇u) = 0.
n→∞
(3.9)
(3.10)
Ω
This result and the following inequality [23, Lemma 2.2]
r−2
r−2
|ξ|
ξ − |η|
η (ξ − η) ≥ 2−r |ξ − η| , ∀r ≥ 2; ξ, η ∈ RN .
(3.11)
Z
(3.12)
yield
lim
n→∞
p(x)
|∇un − ∇u|
dx = 0.
Ω
This fact and Proposition 2.3 imply k∇un − ∇ukp(x) → 0 as n → ∞. Relation (3.12) and fact that
1,p(x)
un ⇀ u (weakly) in W0
(Ω) enable us to apply [12] in order to obtain that un ⇀ u (strongly)
1,p(x)
in W0
(Ω). Thus, Lemma 3.2 is proved.
Now, we show that the Mountain-Pass theorem can be applied in this case.
Lemma 3.3 Assume p, q, h ∈ C+ (Ω) and condition (1.2) is fulfilled. The following assertions
hold.
(i) There exist λ > 0, α > 0 and ρ ∈ (0, 1) such that
1,p(x)
Jλ (u) ≥ α, ∀u ∈ W0
(ii) There exists ω ∈
1,p(x)
W0
(Ω)
(Ω) with kuk = ρ.
such that
lim Jλ (tω) = −∞.
t→∞
1,p(x)
(iii) There exists ϕ ∈ W0
(3.13)
(3.14)
(Ω) such that ϕ ≥ 0, ϕ 6= 0 and
Jλ (tϕ) < 0,
(3.15)
EJQTDE, 2010 No. 65, p. 8
for t > 0 small enough.
Proof. (i) Using Propositions 2.1, 2.2, 2.6 (ii) and Lemma 2.4 we deduce that for any u ∈
(Ω) with ρ ∈ (0, 1) we have
Z
Z
Z
λ
λ
1
p(x)
q(x)
h(x)
|∇u|
dx −
a(x) |u|
dx −
b(x) |u|
dx
Jλ (u) ≥
p2
q1
h1
1,p(x)
W0
Ω
≥
≥
≥
Ω
Ω
λ
1
λ
q(x)
h(x)
p
kuk 2 −
kakβ(x)
|u|
kbkγ(x)
|u|
p(x) −
p∗ (x)
p2
q1
h1
q(x)
h(x)
λ
λ
1
p2
q1
h1
kuk −
kakβ(x) kukp(x) −
kbkγ(x) kukp∗ (x)
p2
q1
h1
λ
λ
1
kukp2 − C1q1 kakβ(x) kukq1 − C1h1
kbkγ(x) kukh1 .
p2
q1
h1
Taking
λ = min
we obtain
Jλ (u) ≥
(
h1 ρp2 −h1
q1 ρp2 −q1
,
q1
4C1 p2 kakβ(x) 4C1h1 p2 kbkγ(x)
)
ρp2
1,p(x)
, ∀u ∈ W0
(Ω) with kuk = ρ.
2p2
Thus Lemma 3.3 (i) is proved.
(ii) Let ω ∈ C0∞ (Ω) , ω ≥ 0, ω 6= 0 and t > 1.We have
Jλ (tω)
=
≤
Z q(x)
Z h(x)
t
t
tp(x)
p(x)
q(x)
h(x)
|∇ω|
dx − λ
a(x) |ω|
dx − λ
b(x) |ω|
dx
p(x)
q(x)
h(x)
Ω
Ω
Ω
Z
Z
Z
tp2
tq2
th 2
p(x)
q(x)
h(x)
|∇ω|
dx − λ
a(x) |ω|
dx − λ
b(x) |ω|
dx.
p1
q1
h1
Z
Ω
Ω
Ω
Since q2 , p2 < h2 we have Jλ (tω) → −∞. Thus Lemma 3.3 (ii) is proved.
(iii) Let ϕ ∈ C0∞ (Ω) , ϕ ≥ 0, ϕ 6= 0 and t ∈ (0, 1). We have
Z
Z
Z
tp1
tq1
th 1
p(x)
q(x)
h(x)
Jλ (tϕ) ≤
|∇ϕ|
dx − λ
a(x) |ϕ|
dx − λ
b(x) |ϕ|
dx.
p1
q1
h1
Ω
Ω
Ω
Since
λtq1
q1
<
λtq1
q1
Z
a(x) |ϕ|q(x) dx +
Ω
λth1
h1
Z
b(x) |ϕ|h(x) dx
Ω
Z
Z
a(x) |ϕ|q(x) dx + b(x) |ϕ|h(x) dx ,
Ω
Ω
EJQTDE, 2010 No. 65, p. 9
we have
tp1
Jλ (tϕ) ≤
p1
for t < δ
1
p1 −q1
with
Z
Ω
Z
Z
q1
λt
p(x)
a(x) |ϕ|q(x) dx + b(x) |ϕ|h(x) dx < 0,
|∇ϕ|
dx −
q1
Ω
Ω
Z
Z
q(x)
h(x)
λp1 a(x) |ϕ|
dx + b(x) |ϕ|
dx
Ω
Ω
Z
.
0 < δ < min 1,
p(x)
q
|∇ϕ|
dx
1
Ω
Lemma 3.3 (iii) is proved.
Proof of Theorem 3.1 We set
n
o
1,p(x)
G = g ∈ C([0, 1] , W0
(Ω)) : g (0) = 0, g (1) = υ ,
1,p(x)
where υ ∈ W0
(Ω) is determined by Lemma 3.3 (ii) and (iii), and
β := inf sup Jλ (g (t)) .
g∈G t∈[0,1]
According to Lemma 3.3 (ii) and (iii), we know that kυk > ρ, so every path g ∈ G intersects
the sphere kvk = ρ. Then Lemma 3.3 (i) implies
α ≤ inf Jλ (u) ≤ β,
kuk=ρ
with the constant α > 0 in Lemma 3.3 (i), thus β > 0. By the Mountain-Pass theorem Jλ admits
a critical value β ≥
α.
1,p(x)
(Ω) , R , from Lemma 3.2 we conclude
Since Jλ ∈ C 1 W0
Jλ′ (un ) → Jλ′ (u)
(3.16)
as n → ∞.
Relations (3.3), (3, 4) and (3.16) show that Jλ′ (u) = 0 and thus u is a weak solutions for problem
(Pλ ). Moreover, by relations (3.3), (3, 4) it follows that Jλ (u) > 0 and thus, u is a nontrivial weak
solutions for problem (Pλ ). The proof is completed.
Remark 3.4 If (u, λ) is a solution of (Pλ ) and u 6= 0, as usual, we call λ and u an eigenvalue
eigenfunction corresponding to λ of (Pλ ), respectively. If (u, λ) is a solution of (Pλ ) and u 6= 0,
then
Z
p(x)
|∇u|
dx
λ = λ(u) = Z
Ω
q(x)
(a(x) |u|
h(x)
+ b(x) |u|
)dx
Ω
EJQTDE, 2010 No. 65, p. 10
and hence λ > 0.
Theorem 3.1 ensures that problem (Pλ ) has a continuous family of positive eigenvalues that lie
in a neighborhood of the origin. Furthermore, we obtain
Z
p(x)
|∇u|
dx
inf
Ω
1,p(x)
u∈W0
\{0}
Z
q(x)
(a(x) |u|
h(x)
+ b(x) |u|
= 0.
)dx
Ω
Remark 3.5 Furthermore, we can consider the equation
(
− div |∇u|p(x)−2 ∇u = λ(a (x) |u|q(x)−2 u − b(x) |u|h(x)−2 u), for x ∈ Ω,
u = 0,
for x ∈ ∂Ω.
/
(Pλ )
where p, q, h, a, b and λ are the same in problem (Pλ ).
/
1,p(x)
The energy functional corresponding to problem (Pλ ) is defined as Iλ : W0
(Ω) → R,
Iλ (u) =
|∇u|p(x)
dx − λ
p(x)
Z
Z
a(x) q(x)
|u|
dx + λ
q(x)
Ω
Ω
We infer that for any x ∈ Ω and u ∈
≤
≤
≤
1,p(x)
W0
Z
b(x)
|u|h(x) dx.
h(x)
Ω
(Ω)
λb(x)
λa(x)
q(x)
h(x)
|u(x)|
−
|u(x)|
q(x)
h(x)
λa2
λb1
|u(x)|q(x) −
|u(x)|h(x)
q1
h2
q(x)
λa2 a2 h2 h(x)−q(x)
q1
b1 q1
"
q2
q1 #
λa2
a2 h2 h2 −q1
a2 h2 h1 −q2
:= A
+
q1
b1 q1
b1 q1
(3.17)
where A is a positive constant independent of u and x, by using the following elementary inequality
[19, Lemma 4]
atk − bts ≤ a
k
a s−k
b
,
for all t ≥ 0
where a, b > 0 and 0 < k < s.
Integrating (3.17) over Ω, we get
Z
Z
a(x)
b(x)
q(x)
h(x)
λ
|u(x)|
dx − λ
|u(x)|
dx ≤ D
q(x)
h(x)
Ω
Ω
where D is a positive constant independent of u. Thus,
EJQTDE, 2010 No. 65, p. 11
Jλ (u) ≥
≥
1
p2
Z
p(x)
|∇u|
Ω
dx − λ
Z
Ω
a(x) q(x)
|u|
dx + λ
q(x)
Z
b(x) h(x)
|u|
dx
h(x)
Ω
1
p
kuk 1 − D,
p2
1,p(x)
for all u ∈ W0
(Ω) with kuk > 1. We infer that Jλ (u) → ∞ as kuk → ∞. Therefore the energy
1,p(x)
functional Iλ is coercive on W0
(Ω). Moreover, a similar argument as the one used in the proof
1,p(x)
(Ω). These facts
of [16, Lemma 3.4] shows that Iλ is also weakly lower semi-continuous in W0
1,p(x)
enable us to apply [22, Theorem 1.2] in order to find that there exits uλ ∈ W0
(Ω) a global
/
minimizer of Iλ and thus, a weak solution of problem (Pλ ).
Acknowledgments
The authors would like to thank the referees and Professor Paul Eloe for their many helpful
suggestions and corrections, which improve the presentation of this paper.
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