Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 120, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
SOLUTIONS OF p(x)-LAPLACIAN EQUATIONS WITH CRITICAL
EXPONENT AND PERTURBATIONS IN RN
XIA ZHANG, YONGQIANG FU
Abstract. Based on the theory of variable exponent Sobolev spaces, we study
a class of p(x)-Laplacian equations in RN involving the critical exponent.
Firstly, we modify the principle of concentration compactness in W 1,p(x) (RN )
and obtain a new type of Sobolev inequalities involving the atoms. Then, by
using variational method, we obtain the existence of weak solutions when the
perturbation is small enough.
1. Introduction
We study the solutions to the problem
∗
− div(|∇u|p(x)−2 ∇u) + |u|p(x)−2 u = |u|p
where p is Lipschitz continuous on R
N
(x)−2
u + h(x),
x ∈ RN ,
(1.1)
and satisfies
1 < p− ≤ p(x) ≤ p+ < N,
(1.2)
0
0 ≤ h(6≡ 0) ∈ Lp (x) (RN ).
We will study (1.1) in the frame of variable exponent function spaces, the definitions of which will be given in section 2.
We say that u ∈ W 1,p(x) (RN ) is a weak solution of problem (1.1), if for any
v ∈ W 1,p(x) (RN ),
Z
∗
|∇u|p(x)−2 ∇u∇v + |u|p(x)−2 uv − |u|p (x)−2 uv − h(x)v dx = 0.
RN
We can verify that the weak solution for (1.1) coincide with the critical point of
the energy functional on W 1,p(x) (RN ):
Z
∗
|∇u|p(x) + |u|p(x)
|u|p (x)
(
− ∗
− h(x)u) dx.
ϕ(u) =
p(x)
p (x)
RN
If h(x) ≡ 0, it is easy to verify that u = 0 is a trivial solution to (1.1). The
existence of nontrivial weak solutions for a class of p(x)-Laplacian equations without
perturbations was studied in [3, 10, 12, 19] via variational methods. They verified
2000 Mathematics Subject Classification. 35J60, 46E35.
Key words and phrases. Variable exponent Sobolev space; critical exponent; weak solution.
c
2012
Texas State University - San Marcos.
Submitted June 19, 2012. Published July 19, 2012.
Supported by grants HIT.NSRIF.2011005 from the Fundamental Research Funds for
the Central Universities, and BK21 from POSTECH..
1
2
X. ZHANG, Y. FU
EJDE-2012/120
the Palais-Smale conditions for the energy functional ϕ and obtained critical points
for ϕ. Moreover, they obtained weak solutions for the p(x)-Laplacian equations.
In [12], we study the following type of p(x)-Laplacian equations with critical
exponent:
∗
− div(|∇u|p(x)−2 ∇u) + λ|u|p(x)−2 u = f (x, u) + h(x)|u|p
(x)−2
u,
x ∈ RN . (1.3)
The difficulty is due to the loss of compactness for the embedding W 1,p(x) (RN ) ,→
∗
Lp (x) (RN ). To prove the Palais-Smale condition for the corresponding energy functional, we assume that the coefficient h(x) of critical part satisfies h(0) = h(∞) = 0.
Then, based on the principle of concentration compactness on W 1,p(x) (RN ) and
symmetric critical point theorem, we obtain infinitely many radial weak solutions
for (1.3).
When p(x) is constant, equations with critical growth have been studied extensively, see for example [2, 5, 14, 21, 22]. The aim of this paper is to use variational
method to show that (1.1) has at least one weak solution if p(x) is function and
h(x) 6≡ 0. Here the difficulty is also caused by the loss of the compactness for the
∗
embedding W 1,p(x) (RN ) ,→ Lp (x) (RN ). In this paper, by using Ekeland’s variational principle [9], we obtain a Palais-Smale sequence if khkp0 (x) is sufficient small.
We do not expect to prove the Palais-Smale condition for ϕ and will not make
similar assumptions as in [12]. However, based on the principle of concentration
compactness on variable exponent Sobolev space established in [12], we prove that
the weak limit of Palais-Smale sequence is a weak solution for (1.1) (see Theorem
3.3). In order to obtain the main result, we also give a kind of modified Sobolev
inequalities involving the atoms in the concentration-compactness principle (see
Theorem 2.7) .
2. Preliminaries
In the studies of nonlinear problems with variable exponential growth, see for
example [1, 3, 4, 6, 10, 15, 16, 20], variable exponent spaces play an important
role. Since they were thoroughly studied by Kovác̆ik and Rákosnı́k [13], variable
exponent spaces have been used to model various phenomena. In [17], Růz̆ic̆ka
presented the mathematical theory for the application of variable exponent Sobolev
spaces in electro-rheological fluids. As another application, Chen, Levine and Rao
[7] suggested a model for image restoration based on a variable exponent Laplacian.
For the convenience of the reader, we recall some definitions and basic properties
of variable exponent spaces Lp(x) (Ω) and W 1,p(x) (Ω), where Ω ⊂ RN is a domain.
For a deeper treatment on these spaces, we refer to [8].
Let P(Ω) be the set of all Lebesgue measurable functions p : Ω → [1, ∞], we
denote
Z
|u|p(x) dx + sup |u(x)|,
ρp(x) (u) =
Ω\Ω∞
x∈Ω∞
where Ω∞ = {x ∈ Ω : p(x) = ∞}.
The variable exponent Lebesgue space Lp(x) (Ω) is the class of all functions u
such that ρp(x) (tu) < ∞, for some t > 0. Lp(x) (Ω) is a Banach space equipped with
the norm
u
kukp(x) = inf{λ > 0 : ρp(x) ( ) ≤ 1}.
λ
EJDE-2012/120
SOLUTIONS OF p(x)-LAPLACIAN EQUATIONS
For any p ∈ P(Ω), we define the


∞,
0
p (x) = 1,

 p(x) ,
p(x)−1
3
conjugate function p0 (x) as
x ∈ Ω1 = {x ∈ Ω : p(x) = 1},
x ∈ Ω∞ ,
x ∈ Ω \ (Ω1 ∪ Ω∞ ).
0
Theorem 2.1. Let p ∈ P(Ω). For any u ∈ Lp(x) (Ω) and v ∈ Lp (x) (Ω),
Z
|uv| dx ≤ 2kukp(x) kvkp0 (x) .
Ω
For any p ∈ P(Ω), we denote
p+ = sup p(x),
p− = inf p(x)
x∈Ω
x∈Ω
and denote by p1 p2 the fact that inf x∈Ω (p2 (x) − p1 (x)) > 0.
Theorem 2.2. Let p ∈ P(Ω) with p+ < ∞. For any u ∈ Lp(x) (Ω), we have
R
p+
p−
;
(1) if kukp(x) ≥ 1, then kukp(x)
≤ Ω |u|p(x) dx ≤ kukp(x)
R
p−
p+
.
(2) if kukp(x) < 1, then kukp(x)
≤ Ω |u|p(x) dx ≤ kukp(x)
The variable exponent Sobolev space W 1,p(x) (Ω) is the class of all functions
u ∈ Lp(x) (Ω) such that |∇u| ∈ Lp(x) (Ω). W 1,p(x) (Ω) is a Banach space equipped
with the norm
kuk1,p(x) = kukp(x) + k∇ukp(x) .
1,p(x)
(Ω) we denote the subspace of W 1,p(x) (Ω) which is the closure of
By W0
∞
C0 (Ω) with respect to the norm k · k1,p(x) . Under the condition 1 ≤ p− ≤ p(x) ≤
1,p(x)
(Ω) are reflexive. And we denote the dual space
p+ < ∞, W 1,p(x) (Ω) and W0
1,p(x)
−1,p0 (x)
(Ω) by W
(Ω).
of W0
For u ∈ W 1,p(x) (Ω), if we define
Z
|u|p(x) + |∇u|p(x)
k|uk| = inf{t > 0 :
dx ≤ 1},
tp(x)
Ω
then k| · k| and k · k1,p(x) are equivalent norms on W 1,p(x) (Ω). In fact, we have
1
kuk1,p(x) ≤ k|uk| ≤ 2kuk1,p(x) .
2
Theorem 2.3. For any u ∈ W 1,p(x) (Ω), we have
R
(1) if k|uk| ≥ 1, then k|uk|p− ≤ Ω (|∇u|p(x) + |u|p(x) ) dx ≤ k|uk|p+ ;
R
(2) if k|uk| < 1, then k|uk|p+ ≤ Ω (|∇u|p(x) + |u|p(x) ) dx ≤ k|uk|p− .
Theorem 2.4. Let Ω be a bounded domain with the cone property. If p ∈ C(Ω̄)
satisfying (1.2) and q is a measurable function defined on Ω with
p(x) ≤ q(x) p∗ (x) ,
N p(x)
N − p(x)
a.e. x ∈ Ω,
then there is a compact embedding W 1,p(x) (Ω) ,→ Lq(x) (Ω).
Theorem 2.5. Let Ω be a domain with the cone property. If p is Lipschitz continuous and satisfies (1.2), q is a measurable function defined on Ω with
p(x) ≤ q(x) ≤ p∗ (x)
then there is a continuous embedding W
1,p(x)
a.e. x ∈ Ω,
(Ω) ,→ Lq(x) (Ω).
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X. ZHANG, Y. FU
EJDE-2012/120
In the proof of main results in Section 3, we will use the following principle of
concentration compactness in W 1,p(x) (RN ) established in [12].
Theorem 2.6. Let {un } ⊂ W 1,p(x) (RN ) with k|un k| ≤ 1 such that
un → u
weakly in W 1,p(x) (RN ),
|∇un |p(x) + |un |p(x) → µ
∗
|un |p
(x)
weak-∗ in M (RN ),
weak-∗ in M (RN ),
→ν
as n → ∞. Denote
C ∗ = sup{
Z
∗
|u|p
(x)
dx : k|uk| ≤ 1, u ∈ W 1,p(x) (RN )}.
RN
Then the limit measures are of the form
X
µ = |∇u|p(x) + |u|p(x) +
µj δxj + µ
e,
µ(RN ) ≤ 1,
j∈J
p∗ (x)
ν = |u|
+
X
νj δxj ,
ν(RN ) ≤ C ∗ ,
j∈J
where J is a countable set, {µj }, {νj } ⊂ [0, ∞), {xj } ⊂ RN , µ
e ∈ M (RN ) is a nonatomic nonnegative measure. The atoms and the regular part satisfy the generalized
Sobolev inequality
∗
∗
∗
ν(RN ) ≤ 2(p+ p+ )/p− C ∗ max{µ(RN )p+ /p− , µ(RN )p− /p+ },
p∗
+
p−
∗
p∗ /p+
νj ≤ C max{µj , µj −
(2.1)
},
where p∗+ = supx∈RN p∗ (x), p∗− = inf x∈RN p∗ (x).
To obtain the main result, we prove the following modified version of Theorem
2.6 in which we give a new form of the inequality (2.1).
Theorem 2.7. Under the hypotheses of Theorem 2.6, for any j ∈ J, the atom xj
satisfies:
∗
p∗ (xj )
p(xj )
νj ≤ C µj
,
(2.2)
where J and xj are as in Theorem 2.6.
Firstly, we give two lemmas.
Lemma 2.8. Let x ∈ RN . For any δ > 0, there exists k(δ) > 0 independent of x
r
r
such that for 0 < r < R with Rr ≤ k(δ), there is a cut-off function ηR
with ηR
≡1
r
1,p(x)
N
in Br (x), ηR ≡ 0 outside BR (x), and for any u ∈ W
(R ),
Z
r
r
(|∇(ηR
u)|p(x) + |ηR
u|p(x) ) dx
BR (x)
Z
≤
(|∇u|p(x) + |u|p(x) ) dx + δ max{k|uk|p+ , k|ukp− }.
BR (x)
The above lemma is obtained by a similar discussion to the one in [11, Lemma
3.1].
EJDE-2012/120
SOLUTIONS OF p(x)-LAPLACIAN EQUATIONS
5
Lemma 2.9. Let x ∈ RN , δ > 0 and Rr < k(δ), where k(δ) is from Lemma 2.8.
Then for any u ∈ W 1,p(x) (RN ), we have
Z
∗
|u|p (x) dx
Br (x)
≤ C ∗ max
n Z
(|∇u|p(x) + |u|p(x) ) dx + δ max{k|uk|p+ , k|uk|p− }
p∗x,R,+ /px,R,−
,
BR (x)
Z
|∇u|p(x) + |u|p(x) ) dx + δ max{k|uk|p+ , k|uk|p− }
p∗x,R,− /px,R,+ o
,
BR (x)
where
px,R,− ,
p∗x,R,− ,
inf
p(y),
y∈BR (x)
sup
p(y),
y∈BR (x)
p∗ (y),
inf
px,R,+ ,
y∈BR (x)
p∗x,R,+ ,
p∗ (y).
sup
y∈BR (x)
r
in Lemma 2.8 and the definition of C ∗ , we
Proof. Using the cut-off function ηR
obtain
Z
Z
∗
r p∗ (x)
|u|p (x) dx ≤
|uηR
|
dx
Br (x)
BR (x)
∗
∗
r px,R,+
r px,R,−
≤ C ∗ max{k|uηR
k|
, k|uηR
k|
}
Z
n
p∗x,R,+ /px,R,−
r p(x)
r p(x)
≤ C ∗ max
(|∇(uηR
)|
+ |uηR
|
) dx
,
BR (x)
Z
r p(x)
r p(x)
(|∇(uηR
)|
+ |uηR
|
) dx
p∗x,R,− /px,R,+ o
.
BR (x)
Then, by Lemma 2.8, we obtain the result.
Proof of Theorem 2.7. Let x0 ∈ RN . By Lemma 2.9, for any δ > 0, there exists
k(δ) > 0 such that for 0 < r < R with r/R ≤ k(δ),
Z
∗
|un |p (x) dx
Br (x0 )
n Z
≤ C ∗ max
(|∇un |p(x) + |un |p(x) ) dx
BR (x0 )
p∗x ,R,+ /px0 ,R,−
0
+ δ max{k|un k|p+ , k|un k|p− }
,
Z
p∗x ,R,− /px0 ,R,+ o
0
(|∇un |p(x) + |un |p(x) ) dx + δ max{k|un k|p+ , k|un k|p− }
.
BR (x0 )
For any 0 < r0 < r, R0 > R. Let η1 ∈ C0∞ (Br (x0 )) such that 0 ≤ η1 ≤ 1; η1 ≡ 1
in Br0 (x0 ), η2 ∈ C0∞ (BR0 (x0 )) such that 0 ≤ η2 ≤ 1; η2 ≡ 1 in BR (x0 ). We obtain
Z
∗
|un |p (x) η1 dx
N
R
Z
∗
≤
|un |p (x) dx
Br (x0 )
≤ C ∗ max
n Z
BR (x0 )
(|∇un |p(x) + |un |p(x) ) dx + δ
p∗x
0 ,R,+
/px0 ,R,−
,
6
X. ZHANG, Y. FU
Z
EJDE-2012/120
(|∇un |p(x) + |un |p(x) ) dx + δ
p∗x
0 ,R,−
/px0 ,R,+ o
.
BR (x0 )
Letting n → ∞, we obtain
ν(B̄r0 (x0 ))
Z
≤
η1 dν
RN
≤ C ∗ max
n Z
η2 dµ + δ
p∗x
0 ,R,+
/px0 ,R,−
,
Z
RN
η2 dµ + δ
p∗x
0 ,R,−
/px0 ,R,+ o
.
RN
Thus
ν({x0 })
≤ ν(B̄r0 (x0 ))
n
p∗x ,R,− /px0 ,R,+ o
p∗
/px ,R,−
0
≤ C ∗ max µ(B̄R0 (x0 )) + δ x0 ,R,+ 0
, µ(B̄R0 (x0 )) + δ
,
where B̄R0 (x0 ) is the closure of BR0 (x0 ). Let δ → 0, R0 → 0. Thus we have
n
o
∗
∗
ν({x0 }) ≤ C ∗ max µ({x0 })p (x0 )/p(x0 ) , µ({x0 })p (x0 )/p(x0 )
∗
= C ∗ µ({x0 })p
(x0 )/p(x0 )
.
p∗ (xj )/p(xj )
Then, for any j ∈ J, the atom xj satisfies νj ≤ C ∗ µj
complete.
. The proof is
3. Main Results
In this section, we prove that (1.1) has at least one nontrivial weak solution
u0 ∈ W 1,p(x) (RN ). First, we prove the following preliminary result which will show
that the weak limit of Palais-Smale sequence of ϕ is a weak solution for (1.1) (see
Theorem 3.3).
Throughout this paper, we denote by C universal positive constants unless otherwise specified.
Theorem 3.1. Let {un } be a sequence in W 1,p(x) (RN ) such that un → u weakly
0
in W 1,p(x) (RN ) and ϕ0 (un ) → 0 in W −1,p (x) (RN ), as n → ∞. Then ∇un → ∇u
N
0
a.e. in R , as n → ∞. Moreover, ϕ (u) = 0 .
Proof. Since un → u weakly in W 1,p(x) (RN ), passing to a subsequence, still denoted
by {un }, we may assume that there exist µ, ν ∈ M (RN ) such that |∇un |p(x) +
∗
|un |p(x) → µ and |un |p (x) → ν weakly-∗ in M (RN ), where M (RN ) is the space of
finite nonnegative Borel measures on RN . By Theorems 2.6 and 2.7, there exist
some countable set J, {µj }, {νj } ⊂ (0, ∞) and {xj } ⊂ RN such that
X
µ = |∇u|p(x) + |u|p(x) +
µj δxj + µ
e,
(3.1)
j∈J
∗
ν = |u|p
(x)
+
X
νj δxj ,
(3.2)
j∈J
p∗ (xj )/p(xj )
νj ≤ C ∗ µj
,
(3.3)
EJDE-2012/120
SOLUTIONS OF p(x)-LAPLACIAN EQUATIONS
where
∗
C = sup
Z
∗
|u|p
(x)
7
dx : k|uk| ≤ 1, u ∈ W 1,p(x) (RN ) ,
RN
where µ
e ∈ M (RN ) is a nonatomic positive measure, δxj is the Dirac measure at xj .
In the following, we prove that J is a finite set or empty. In fact, for any ε > 0,
let φ ∈ C0∞ (B2ε (0)) such that 0 ≤ φ ≤ 1, |∇φ| ≤ 2ε ; φ ≡ 1 on Bε (0). For any j ∈ J,
{φ(·−xj )un } is bounded on W 1,p(x) (RN ). Then we have hϕ0 (un ), φ(·−xj )un → 0,
as n → ∞. Note that
hϕ0 (un ), φ(· − xj )un
Z
∗
|∇un |p(x)−2 ∇un ∇(un φ(x − xj )) + |un |p(x) φ(x − xj ) − |un |p (x) φ(x − xj )
=
RN
− h(x)un φ(x − xj ) dx
Z
(|∇un |p(x) + |un |p(x) )φ(x − xj ) + |∇un |p(x)−2 ∇un ∇φ(x − xj ) · un
=
RN
∗
− |un |p (x) φ(x − xj ) − h(x)un φ(x − xj ) dx.
0
As un → u in Lp(x) (B2ε (xj )) and h ∈ Lp (x) (RN ), we obtain
Z
Z
h(x)un φ(x − xj ) dx →
h(x)uφ(x − xj ) dx,
RN
RN
as n → ∞. Using (3.1) and (3.2) we obtain
Z
lim
|∇un |p(x)−2 ∇un ∇φ(x − xj ) · un dx
n→∞ RN
Z
Z
Z
=
−φ(x − xj ) dµ +
h(x)uφ(x − xj ) dx +
RN
RN
(3.4)
φ(x − xj ) dν.
RN
It is easy to verify that k∇φ(x − xj ) · un kp(x) → k∇φ(x − xj ) · ukp(x) , as n → ∞.
Then
Z
lim |
|∇un |p(x)−2 ∇un ∇φ(x − xj ) · un dx|
n→∞
RN
Z
≤ lim sup
|∇un |p(x)−1 |∇φ(x − xj ) · un | dx
n→∞
RN
≤ lim sup 2k|∇un kp(x)−1 |p0 (x) · k∇φ(x − xj ) · un kp(x) ≤ Ck∇φ(x − xj ) · ukp(x) .
n→∞
Note that
Z
|∇φ(x − xj ) · u|p(x) dx
RN
Z
|∇φ(x − xj ) · u|p(x) dx
=
B2ε (xj )
≤ 2k|∇φ(x − xj )|p(x) k( p∗ (x) )0 ,B2ε (xj ) · k|u|p(x) k p∗ (x) ,B2ε (xj )
p(x)
and
Z
p(x) (
(|∇φ(x − xj )|
)
p∗ (x) 0
)
p(x)
p(x)
Z
dx =
B2ε (xj )
=
2
|∇φ|N dx ≤ ( )N meas(B2ε (xj ))
ε
B2ε (xj )
4N
ωN ,
N
8
X. ZHANG, Y. FU
EJDE-2012/120
p∗ (x)
R
where ωN is the surface area of the unit sphere in RN . As B2ε (xj ) (|u|p(x) ) p(x) dx →
0, as ε → 0, we obtain k∇φ(x − xj ) · ukp(x) → 0, which implies
Z
lim
|∇un |p(x)−2 ∇un ∇φ(x − xj ) · un dx → 0,
n→∞
RN
as ε → 0. Similarly, we can also get
Z
Z
|
h(x)uφ(x − xj ) dx| ≤
RN
|h(x)u| dx → 0,
B2ε (xj )
as ε → 0.
Thus, it follows from (3.4) that 0 = −µ({xj }) + ν({xj }); i.e., µj = νj for any
j ∈ J. Using (3.3) we obtain
p∗ (xj )/p(xj )
νj ≤ C ∗ µj
p(xj )
p(xj )−p∗ (xj )
,
−
p−
−
p+
≥ min{(C ∗ ) (p∗ −p)+ , (C ∗ ) (p∗ −p)− } for
which implies that νj ≥ (C ∗ )
any j ∈ J. As ν is finite, J must be a finite set or empty.
Next, we prove that ∇un → ∇u a.e. in RN , as n → ∞.
(1) If J is a finite nonempty set, say J = {1, 2, . . . , m}. Let d = min{d(xi , xj ) :
i, j ∈ J with i 6= j}. There exists R0 > 0 such that Bd (xj ) ⊂ BR0 for any j ∈ J.
Take 0 < ε < d4 , B2ε (xi ) ∩ B2ε (xj ) = ∅ for any i, j ∈ J with i 6= j. Denote
ΩR,ε = {x ∈ BR : d(x, xj ) > 2ε for any j ∈ J}.
In the following, we will verify that for any R > R0 ,
Z
(|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u) dx → 0, as n → ∞.
ΩR,ε
Let ψ ∈ C0∞ (B2R ) such that 0 ≤ ψ ≤ 1; ψ ≡ 1 on BR . Define
ψε (x) = ψ(x) −
m
X
φ(x − xj ).
j=1
We derive that ψε ∈ C0∞ (B2R ) such that 0 ≤ ψε ≤ 1; ψε ≡ 0 on ∪m
j=1 Bε (xj ) and
ψε ≡ 1 on (RN \ ∪m
j=1 B2ε (xj )) ∩ BR . Thus
Z
0≤
(|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u) dx
ΩR,ε
Z
(|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u)ψε dx
Z
0
0
= hϕ (un ), un ψε i − hϕ (un ), uψε i −
|∇u|p(x)−2 ∇u(∇un − ∇u)ψε dx
B2R
Z
∗
−
|∇un |p(x)−2 ∇un ∇ψε · un + |un |p(x) ψε − |un |p (x) ψε − h(x)un ψε dx
B
Z 2R +
|∇un |p(x)−2 ∇un ∇ψε · u + |un |p(x)−2 un uψε
B2R
∗
− |un |p (x)−2 un uψε − h(x)uψε dx.
≤
B2R
Note that
Z
|
B2R
(|∇un |p(x)−2 ∇un ∇ψε · un − |∇un |p(x)−2 ∇un ∇ψε · u) dx|
EJDE-2012/120
SOLUTIONS OF p(x)-LAPLACIAN EQUATIONS
Z
9
|∇un |p(x)−1 |un − u| dx
≤C
B2R
≤ Ck|∇un |p(x)−1 kp0 (x) kun − ukp(x),B2R ,
which implies
Z
Z
|∇un |p(x)−2 ∇un ∇ψε · un dx −
B2R
|∇un |p(x)−2 ∇un ∇ψε · u dx → 0,
B2R
as n → ∞. Similarly, we obtain
Z
Z
p(x)
|un |
ψε dx −
B2R
and
|un |p(x)−2 un uψε dx → 0,
B2R
Z
Z
h(x)un ψε dx −
B2R
h(x)uψε dx → 0.
B2R
As un → u weakly in W 1,p(x) (RN ). Using Theorem 2.4 we obtain un → u in
L
(B2R ), for any R > 0. Passing to a subsequence, still denoted by {un }, a
diagonal process enables us to assume that un → u a.e. in RN , as n → ∞. Thus
∗
∗
∗
∗
∗
∗
|un ψε |p (x) → |uψε |p (x) a.e. in RN . As |un − u|p (x) ≤ 2p+ (|un |p (x) + |u|p (x) ), by
Fatou’s Lemma, we have
Z
∗
∗
2p+ +1 |uψε |p (x) dx
N
R
Z
∗
∗
∗
∗
∗
=
lim inf (2p+ |un ψε |p (x) + 2p+ |uψε |p (x) − |un ψε − uψε |p (x) ) dx
RN n→∞
Z
∗
∗
∗
∗
∗
≤ lim inf
(2p+ |un ψε |p (x) + 2p+ |uψε |p (x) − |un ψε − uψε |p (x) ) dx
n→∞
RN
Z
Z
∗
∗
p∗
+1
=
2 + |uψε |p (x) dx − lim sup
|un ψε − uψε |p (x) dx.
p(x)
n→∞
RN
Using (3.2), we have
∗
R
RN
|un |p
Z
(x)
∗
|ψε |p
(x)
RN
∗
|un ψε − uψε |p
∗
R
dx →
RN
(x)
|u|p
(x)
∗
|ψε |p
(x)
dx, thus
dx → 0,
RN
as n → ∞. Moreover, we derive
Z
Z
∗
|un |p (x) ψε dx −
B2R
Then
Z
∗
|un |p
(x)−2
un uψε dx → 0.
B2R
(|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u) dx → 0.
ΩR,ε
As in the proof of [6, Theorem 3.1], ΩR,ε is divided into two parts:
Ω1R,ε = {x ∈ ΩR,ε : p(x) < 2},
Ω2R,ε = {x ∈ ΩR,ε : p(x) ≥ 2}.
On Ω1R,ε , we obtain
Z
|∇un − ∇u|p(x) dx
Ω1R,ε
Z
≤C
Ω1R,ε
p(x)
(|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u) 2
10
X. ZHANG, Y. FU
× |∇un |p(x) + |∇u|p(x)
2−p(x)
2
EJDE-2012/120
dx
p(x)
2
≤ Ck (|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u) 2 k p(x)
, Ω1R,ε
× k(|∇un |p(x) + |∇u|p(x) )
2−p(x)
2
2
k 2−p(x)
, Ω1R,ε .
Note that
Z
(|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u) dx
Ω1R,ε
Z
(|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u) dx,
≤
ΩR,ε
which implies
p(x)/2
k2/p(x),Ω1R,ε → 0.
k (|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u)
R
As {un } is bounded in W 1,p(x) (RN ), we obtain Ω1 |∇un − ∇u|p(x) dx → 0, as
R,ε
n → ∞.
On Ω2R,ε , we obtain
Z
|∇un − ∇u|p(x) dx
Ω2R,ε
Z
(|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u) dx → 0,
≤C
Ω2R,ε
as n → ∞. Thus, we obtain
Z
|∇un − ∇u|p(x) dx → 0
ΩR,ε
for any R > R0 , 0 < 2ε < d2 . Moreover, up to a subsequence, we assume that
∇un → ∇u a.e. in RN .
(2) If J is empty. Let ψ ∈ C0∞ (B2R ) such that 0 ≤ ψ ≤ 1; ψ ≡ 1 in BR , we
obtain
Z
0≤
(|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u) dx
B
Z R
≤
(|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u)ψ dx.
B2R
Similarly to (1), we obtain
Z
(|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u)(∇un − ∇u) dx → 0,
BR
as n → ∞, which implies
Z
|∇un − ∇u|p(x) dx → 0,
BR
for any R > 0. Thus, we may assume that ∇un → ∇u a.e. in RN .
0
As {|∇un |p(x)−2 ∇un } is bounded in (Lp (x) (RN ))N and |∇un |p(x)−2 ∇un conp(x)−2
N
verges to |∇u|
∇u a.e. in R , we obtain
|∇un |p(x)−2 ∇un → |∇u|p(x)−2 ∇u
0
weakly in (Lp (x) (RN ))N .
EJDE-2012/120
SOLUTIONS OF p(x)-LAPLACIAN EQUATIONS
11
Similarly, we obtain
|un |p(x)−2 un → |u|p(x)−2 u
0
weakly in Lp (x) (RN )
and
∗
|un |p
(x)−2
∗
un → |u|p
(x)−2
u
∗
weakly in L(p
(x))0
(RN ).
Thus, for any v ∈ C0∞ (RN ), we have
Z
Z
|∇un |p(x)−2 ∇un ∇v →
|∇u|p(x)−2 ∇u∇v dx,
N
N
R
Z
ZR
p(x)−2
|un |
un v →
|u|p(x)−2 uv dx,
RN
RN
Z
Z
∗
p∗ (x)−2
|un |
un v →
|u|p (x)−2 uv dx.
RN
RN
Note that
hϕ0 (un ), vi =
Z
∗
|∇un |p(x)−2 ∇un ∇v + |un |p(x)−2 un v − |un |p
(x)−2
un v − h(x)v dx
RN
0
and ϕ0 (un ) → 0 in W −1,p (x) (RN ), as n → ∞, we obtain
Z
∗
0
hϕ (u), vi =
|∇u|p(x)−2 ∇u∇v + |u|p(x)−2 uv − |u|p (x)−2 uv − h(x)v dx
RN
(3.5)
= 0.
As p is Lipschitz continuous on RN , it follows that p satisfies the weak Lipschitz
condition [18]. Thus, C0∞ (RN ) is dense on W 1,p(x) (RN ). Using (3.5), we obtain
hϕ0 (u), vi = 0,
for any v ∈ W 1,p(x) (RN ); i.e. ϕ0 (u) = 0.
We remark that in the proof of Theorem 3.1, we use the inequality (2.2) in
Theorem 2.7. As p(x) p∗ (x), p∗ (x) − p(x) ≥ (p∗ − p)− > 0 for any x ∈ RN .
Then, we avoided the assumption p∗− > p+ and obtained that the set of atoms J is
empty or finite.
Next, using Theorem 3.1 we prove that there exists a critical point for ϕ. The
following result of the variational functional ϕ is required by using Ekeland’s variational principle.
Lemma 3.2. There exist ρ0 > 0, h0 > 0 such that if khkp0 (x) ≤ h0 , we have
ϕ(u) > 0 for any u ∈ {u ∈ W 1,p(x) (RN ) : k|uk| = ρ0 }.
Proof. For any u ∈ W 1,p(x) (RN ), we obtain
Z ∗
|u|p (x)
|∇u|p(x) + |u|p(x)
−
−
h(x)u
dx
ϕ(u) ≥
p+
(p∗ )−
RN
Z |∇u|p(x) + |u|p(x)
− h(x)u dx
=
2p+
RN
Z ∗
p(x)
|∇u|
+ |u|p(x)
|u|p (x) +
−
dx.
2p+
(p∗ )−
RN
12
X. ZHANG, Y. FU
EJDE-2012/120
As p(x) p∗ (x) and p(x) are Lipschitz continuous on RN , as in the proof of [6,
Theorem 3.1], there exists a sequence of disjoint open N -cubes {Qi }∞
i=1 with side
r > 0 such that RN = ∪∞
Q
,
i
i=1
pi+ , sup p(x) < p∗i− , inf p∗ (x),
x∈Qi
x∈Qi
and p∗i− − pi+ > γ , 12 inf x∈RN (p∗ (x) − p(x)), for i = 1, 2, . . . .
By [8, Corollary 8.3.2], there exists r0 = r0 (r, N, p+ , p− ) > 1 independent of
i ∈ N such that for any v ∈ W 1,p(x) (Qi ), kvkp∗ (x) ≤ r0 k|vk|. Then, for any
u ∈ W 1,p(x) (RN ), we obtain kukp∗ (x),Qi ≤ r0 k|uk|Qi .
If k|uk| ≤ r0−1 , then k|uk|Qi ≤ k|uk| ≤ r0−1 , for any i ∈ N. Thus, kukp∗ (x),Qi ≤ 1.
Using Theorems 2.2 and 2.3 we obtain
Z ∗
∗
∞ Z
|∇u|p(x) + |u|p(x)
X
|u|p (x) |u|p (x) |∇u|p(x) + |u|p(x)
−
dx
=
−
dx
2p+
(p∗ )−
2p+
(p∗ )−
RN
i=1 Qi
≥
≥
p
∞ X
k|uk|Qi+
i
i=1
∞
X
2p+
p
k|uk|Qi+
i
i=1
p∗
r i−
(p∗ )
− 0∗ k|uk|Qi i−
(p )−
2p+
1−
2p+ p∗i−
γ
r
k|uk|
Qi .
(p∗ )− 0
p∗
i−
Denote ρ0 = min{r0−1 , ( (p2p∗ )+− r0 )−1/γ }. If k|uk| ≤ ρ0 , then
Z ∗
|∇u|p(x) | + |u|p(x)
− |u|p (x) dx ≥ 0.
2
RN
We obtain
k|uk|p+
k|uk|p+
− 2khkp0 (x) kukp(x) ≥
− Ckhkp0 (x) k|uk|.
ϕ(u) ≥
2p+
2p+
Thus, it suffices to take khkp0 (x) small enough.
(3.6)
Then, using Ekeland’s variational principle and Lemma 3.2, we obtain a PalaisSmale sequence for ϕ. Based on Theorem 3.1, we have the following result, which
shows that ϕ has a critical if khkp0 (x) is small. Moreover, we obtain a nontrivial
weak solution for (1.1).
Theorem 3.3. If khkp0 (x) ≤ h0 , there exists u0 ∈ {u ∈ W 1,p(x) (RN ) : k|uk| ≤ ρ0 }
such that u0 is a weak solution of (1.1), where ρ0 , h0 are from Lemma 3.2.
Proof. Denote
c1 = inf{ϕ(u) : u ∈ W 1,p(x) (RN ) with k|uk| ≤ ρ0 }.
It follows from (3.6) that
R c1 > −∞. Note that h(x) ≥ 0 and h(x) 6≡ 0, there exists
v ∈ C0∞ (RN ) such that RN h(x)v dx > 0. Take 0 < s < 1, we obtain
Z
∗
|∇sv|p(x) + |sv|p(x)
|sv|p (x)
ϕ(sv) =
− ∗
− h(x)sv dx
p(x)
p (x)
RN
Z
Z
p(x)
p(x)
|∇v|
+ |v|
≤ sp−
dx − s
h(x)v dx.
p(x)
RN
RN
As p− > 1, we have k|svk| < ρ0 and ϕ(sv) < 0, when s is sufficiently small. Thus
c1 < 0.
EJDE-2012/120
SOLUTIONS OF p(x)-LAPLACIAN EQUATIONS
13
By Ekeland’s variational principle, there exists {un } ⊂ {u ∈ W 1,p(x) (RN ) :
k|uk| ≤ ρ0 } such that ϕ(un ) → c1 and
1
ϕ(w) ≥ ϕ(un ) − k|w − un k|,
(3.7)
n
for any w ∈ W 1,p(x) (RN ) with k|wk| ≤ ρ0 .
Since c1 < 0, we assume that ϕ(un ) < 0. It follows from Lemma 3.2 that
0
k|un k| < ρ0 . Using (3.7), we obtain ϕ0 (un ) → 0 in W −1,p (x) (RN ), as n → ∞. As
{un } is bounded in W 1,p(x) (RN ), we assume that un → u0 weakly in W 1,p(x) (RN ),
then k|u0 k| ≤ ρ0 . By Theorem 3.1, we obtain ϕ0 (u0 ) = 0.
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Xia Zhang
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China.
Department of Mathematics, Pohang University of Science and Technology, Pohang,
Korea
E-mail address: piecesummer1984@163.com
Yongqiang Fu
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
E-mail address: fuyqhagd@yahoo.cn