Weighted variable exponent Sobolev estimates for elliptic equations with non-standard growth and measure data

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by  The Anh Bui

WEIGHTED VARIABLE EXPONENT SOBOLEV ESTIMATES FOR ELLIPTIC
EQUATIONS WITH NON-STANDARD GROWTH AND MEASURE DATA
THEANHBUIANDXUANTHINHDUONG
Abstract. Considerthefollowingnonlinearellipticequationofp(x)-Laplaciantypewithnonstandard
growth
diva(Du,x)=µ in Ω,
7 (u=0 on ∂Ω,
1 where Ω is a Reifenberg domain in Rn, µ is a Radon measure defined on Ω with finite total mass and
0 thenonlinearitya:Rn×Rn→Rn ismodeleduponthep(·)-Laplacian.
2 Weprovetheestimates onweighted variable exponentLebesgue spaces forgradients ofsolutions to
this equation in terms of Muckenhoupt–Wheeden type estimates. As a consequence, we obtain some
n new results such as the weighted Lq−Lr regularity (with constants q < r) and estimates on Morrey
a spacesforgradientsofthesolutionstothisnon-linearequation.
J

4

]
P Contents
A
1. Introduction 1
.
h 2. Assumptions and Statement of the results 3
t
a 2.1. Our assumptions 4
m 2.2. Reifenberg flat domains 4
[ 2.3. Statement of the results 5

3. Approximation results 7
1
3.1. Interior Estimates 7
v
2 3.2. Boundary estimates 13
5 4. Weighted regularity estimates 20
9 References 24
0
0
.
1
0 1. Introduction
7
1 Partial differential equations including nonlinear elliptic and parabolic problems with nonstandard
: growthconditionshaverecentlybeenstudiedextensivelybymanymathematiciansastheseequationshave
v
i had a wide range of applications in many fields such as mathematical physics, elastic mechanics, image
X
processingandelectro-rheologicalfluid dynamics. See for example [2, 3, 13, 20, 17, 27, 39, 40, 41, 46, 47]
r and the references therein.
a
In this paper we consider the following nonlinear elliptic equation of p(x)-Laplacian type with non-
standard growth
diva(Du,x)=µ in Ω,
(1)
u=0 on ∂Ω,
(
where Ω is a bounded open domainin Rn and µ is a Radonmeasure defined on Ω with finite total mass.
The nonlinearity a:Rn Rn Rn is modeled upon the p()-Laplacian.
× → ·
Recently a systematic study on nonlinear elliptic of p(x)-Laplacian of type (1) with measure data has
beenreceivedalotofattention. Intheparticularcaseofp-Laplaciantypeequations(i.e. pisindependent
of x), the existence results for the solutions to nonlinear elliptic and parabolic equations with measure
datawereprovedin[8,9,5,10]. Thentheregularityresultsforsolutionsofthoseequationswereobtained
2010 Mathematics Subject Classification. 35B65,35J60, 35J99.
Keywordsandphrases. nonlinearp(x)-Laplaciantypeequation,measuredata,Reifenbergdomain,weightedgeneralized
Lebesguespaces.
1
2 THEANHBUIANDXUANTHINHDUONG
in [38] for the elliptic case and in [37] for the parabolic case. For the generalcase of p(x)-Laplacian type
equation, some interesting results regarding to entropy solutions and very weak solutions were obtained
in [6, 1, 42, 45].
Recall that a weak solution to the problem (1) is a function u W1,p(·)(Ω) such that
∈ 0
a(Du,x) Dϕdx= ϕdµ for all ϕ C∞(Ω).
ˆ · ˆ ∈ 0
Ω Ω
See Section 2 for definition of the variable exponent Sobolev space W1,p(·)(Ω).
0
For x Rn, we define
∈
µ(B (y)) µ(B (x))
M (µ)(x)=sup sup | | r sup| | r
1 rn−1 ∼ rn−1
r>0Br(y)∋x r>0
tobethefirstorderfractionalmaximalfunctionassociatedtothemeasureµ,whereB (z):= y : z y <
r
r istheopenballwithcenterz Rn andradiusr. Itisnotdifficulttoseethatforanonneg{ative| l−oca|lly
fi}nite measure ν in Rn, the max∈imal function M (ν) is dominated by the Riesz potential related to ν.
1
More precisely, we have
dν(y)
M (ν)(x) c I (ν)(x):=c ,x Rn.
1 ≤ n 1 nˆRn x y n−1 ∈
| − |
We note that the problem of getting estimates for the solution via fractional maximal functions and
nonlinear potentials is an interesting topic and has attracted a great deal of attention in recent years.
We now list some of the papers related to this direction.
(i) The first two results appeared in [28, 29], where the authors proved a pointwise potential estimate
for solutions to the quasi elliptic equation via Wolff potentials. Later, in [44], by using a different
approach, the authors extended this result to obtain the pointwise estimates for solutions to non-
homogeneous quasi-linear equations of p-Laplacian type with measure data in terms of Wolff type
nonlinear potentials.
(ii) In the series of works by Mingione and his collaborators, they extended the results in [44] to the
pointwise estimates for the gradient of solutions, instead of the solution, via nonlinear potentials.
More precisely, the pointwise estimate for the gradient of solutions to the degenerate quasilinear
equations of p-Laplacian type was first proved in [34] for the case p = 2. The case p = 2 can be
6
found in [23, 24, 25, 30, 31].
(iii) The gradient estimates for solutions to the equation (1) in terms of variable exponent potentials
wereobtainedin[7,4]correspondingtop() 2andp()>2 1/nbyusingMingioneandDuzzar’s
· ≥ · −
approach. Then, optimal integrability results for solutions of the p(x)-Laplace equation in variable
exponent weak Lebesgue spaces were obtained in [1].
(iv) The regularity results for the solutions to the nonlinear elliptic equation of p(x)-Laplacian type of
the form
diva(Du,x)=div(F p(·)−2F) in Ω,
| |
(u=0 on ∂Ω,
were proved in [13, 14] in the scale of Lebesgue and generalized Lebesgue spaces, respectively.
(v) In [38], the author proved weighted estimates for gradients of solution to the equation (1) in the
particular case of p-Laplaciantype via the maximal operator M .
1
ThemainaimofthispaperistoprovetheweightedLq(·) estimatesforgradientsofthesolutionstothe
equation(1)viathefractionalmaximalfunctionM . Theseestimatesaresimilartothosein[38]asin(v)
1
abovebutweobtaintheestimatesforsolutionsofnonlinearelliptic equationsofp(x)-Laplaciantype and
intermsoftheweightedLq(·). SeeTheorem2.6anditssubsequenceresultsinCorollaries2.8,2.9and2.10.
REGULARITY ESTIMATES FOR ELLIPTIC EQUATIONS WITH NON-STANDARD GROWTH 3
We now give some comments on the technical ingredients used in this paper. In order to prove the
mainresults,weemploythemaximalfunctiontechniquewhichmakesuseofthevariantofVitalicovering
lemma and good λ-inequality. This technique was originated in [15] and was used in various settings.
See for example [12, 38, 36, 37, 11]. However, some major modifications need to be carried out since
the maximal function techniques are not applicable directly to our problem due to the presence of the
variable exponent p(x) which rules out the homogeneity of the equation (1). To overcome this problem,
we make use of the log-Ho¨lder condition of the exponent functions and some subtle localized estimates.
Theorganizationofthe paperisasfollows. InSetion2,wesetuptheassumptionsonthe nonlinearity
a and the underlying domain Ω, and then state the main results. See Theorem 2.6 and its subsequence
results such as Corollaries 2.8, 2.9, 2.10 and Theorem 2.11. In Section 3, we prove some interior and
boundary comparison estimates which play an important role in the sequel. The proofs of the main
results are given in Section 4.
Throughout the paper, we always use C and c to denote positive constants that are independent of
the main parameters involved but whose values may differ from line to line. We write A . B if there
is a universal constant C so that A CB and A B if A . B and B . A. For a,b R we denote
a b=min a,b . We also denote by≤(data) the in∼finitely smallquantity with respectto∈the data,i.e.,
lim∧ {(dat}a)=0. O
data→0
O
2. Assumptions and Statement of the results
We will begin with some notations which will be used frequently in the sequel.
For x Rn and r >0, we denote by B (x):= y Rn : x y <r the open ball with center x
r
• and ra∈dius r in Rn. { ∈ | − | }
We alsodenoteΩ (x)=Ω B (x)and∂ Ω (x)=∂Ω B (x). Ifxis theorigin,wesimply write
r r w r r
• ∩ ∩
B , Ω and ∂ Ω for B (x), Ω (x) and ∂ Ω (x), respectively.
r r w r r r w r
For a measurable function f on a measurable subset E Rn we define
• ⊂
1
f = fdx= fdx.
E E ˆ
E | | E
Wenowrecallsomedefinitionsandbasicpropertiesconcerningthe variableexponentLebesguespaces
in [18]. Let Ω be a subset of Rn. For p():Ω (0, ), we define the variable exponent Lebesgue spaces
· → ∞
Lp(·)(Ω) to be a generalization of the classical Lebesgue spaces consisting of all measurable functions on
Ω satisfying
f(x)p(x)dx< ,
ˆ | | ∞
Ω
with the norm
f(x) p(x)
kfkLp(·)(Ω) =inf λ>0:ˆ | λ | dx≤1 .
n Ω(cid:18) (cid:19) o
It is well-known that
kfkLp(·)(Ω) ≤1⇐⇒ˆ |f(x)|p(x)dx≤1,
Ω
and if 1 ≤ p(x) < ∞ for all x ∈ Ω then k·kLp(·)(Ω) is a norm and hence Lp(·)(Ω) is a Banach space. In
general, k·kLp(·)(Ω) is a quasi-norm.
ThegeneralizedSobolevspaceW1,p(·)(Ω)isdefinedasthesetofallmeasurablefunctionsf Lp(·)(Ω)
∈
whose derivative Df Lp(·)(Ω). If f W1,p(·)(Ω), then its norm is defined by
∈ ∈
kfkW1,p(·)(Ω) =kfkLp(·)(Ω)+k|Df|kLp(·)(Ω).
The space W1,p(·)(Ω) is defined as a closure of C∞(Ω) in W1,p(·)(Ω). The generalizedLebesgue–Sobolev
0 c
spaces play an important role in studying regularity estimates for elliptic and parabolic problems. See
for example [20, 21, 18] and the references therein for further discussions.
4 THEANHBUIANDXUANTHINHDUONG
2.1. Our assumptions. In this paper, we assume that the nonlinearity a(ξ,x) : Rn Rn Rn is
measurable in x for every ξ Rn and differentiable in ξ for each x Rn. In addition,×there →exist the
∈ ∈
variable exponent p():Ω (1, ) and constants Λ Λ >0, s [0,1] so that
1 2
· → ∞ ≥ ∈
(2) (s2+ ξ 2)1/2 Dξa(ξ,x) + a(ξ,x) Λ1(s2+ ξ 2)p(x2)−1,
| | | | | |≤ | |
and
(3) Dξa(ξ,x)η,η Λ2(s2+ ξ 2)p(x2)−2 η 2,
h i≥ | | | |
for every x,ξ,η Rn.
∈
Note that these two conditions imply that
(4) a(ξ,x) a(η,x),ξ η Λ2(s2+ ξ 2+ η 2)p(x2)−2 ξ η 2 Λ2 ξ η p(x)
h − − i≥ | | | | | − | ≥ | − |
for every x,ξ,η Rn.
∈
Moreover,the exponent function p():Ω (1, ) is assumed to be continuous, satisfies the bounds
· → ∞
1
2 <γ p(x) γ < ,
1 2
− n ≤ ≤ ∞
and the log-Ho¨lder continuity condition
(5) p(x) p(y) ω(x y ), x,y Rn
| − |≤ | − | ∀ ∈
where ω :[0, ) [0, ) is a non-decreasing function satisfying
∞ → ∞
1
(6) lim ω(r)log =0.
r→0+ r
(cid:16) (cid:17)
We choose a number R so that for all 0<r<R ,
ω ω
1 1
(7) 0<ω(r)log .
r ≤ 2
(cid:16) (cid:17)
We set
a(ξ,x) a(ξ, )
Θ(a,B (y))(x)= sup ·
r ξ∈Rn(cid:12)(cid:12)(cid:12)(s2+|ξ|2)p(x2)−1 − (s2+|ξ|2)p(·2)−1!Br(y)(cid:12)(cid:12)(cid:12)
which is used in the next definition conce(cid:12)rning the nonlinearity a. (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Definition 2.1. Let R ,δ >0. The nonlinearity a is said to satisfy a small (δ,R )-BMO condition if
0 0
(8) [a] := sup sup Θ(a,B (y))(x)2dx δ2.
2,R0 y∈Rn0<r≤R0 Br(y)| r | ≤
Remark 2.2. This condition was introduced in [13]. Note that if (8) holds true, then for any β [1, )
∈ ∞
we have
[a] := sup sup Θ(a,B (y))(x)βdx (δ).
β,R0 y∈Rn0<r≤R0 Br(y)| r | ≤O
2.2. Reifenbergflatdomains. ConcerningtheunderlyingdomainΩ,wedonotassumeanysmoothness
condition on Ω, but the following flatness condition.
Definition 2.3. Let δ,R >0. The domain Ω is said to be a (δ,R ) Reifenberg flat domain if for every
0 0
x ∂Ω and 0<r R , then there exists a coordinate system depending on x and r, whose variables are
0
∈ ≤
denoted by y =(y ,...,y ) such that in this new coordinate system x is the origin and
1 n
(9) B y :y >δr B Ω y :y > δr .
r n r n
∩{ }⊂ ∩ ⊂{ − }
Remark 2.4. (a)The condition of (δ,R )-Reifenbergflatness condition was first introducedin [40]. This
0
condition does not require any smoothness on the boundary of Ω, but sufficiently flat in the Reifenberg’s
sense. The Reifenberg flat domain includes domains with rough boundaries of fractal nature, and Lips-
chitz domains with small Lipschitz constants. For further discussions about the Reifenberg domain, we
refer to [40, 19, 43] and the references therein.
REGULARITY ESTIMATES FOR ELLIPTIC EQUATIONS WITH NON-STANDARD GROWTH 5
(b) If Ω is a (δ,R ) Reifenberg domain, then for any x ∂Ω and 0 < ρ < R (1 δ) there exists a
0 0 0
∈ −
coordinate system, whose variables are denoted by y =(y ,...,y ) such that in this coordinate system the
1 n
origin is an interior point of Ω, x =(0,...,0, δρ ) and
0 −1−δ
2δρ
B+ B Ω B y :y > .
ρ ⊂ ρ∩ ⊂ ρ∩ n −1 δ
(cid:26) − (cid:27)
(c) For x Ω and 0<r <R , we have
0
∈
B (x) 2 n
r
(10) | | .
B (x) Ω ≤ 1 δ
r
| ∩ | (cid:16) − (cid:17)
Throughout the paper, we always assume that the domain Ω is a (δ,R ) Reifenberg flat domain, and
0
the nonlinearity a satisfies (2), (3) and the small (δ,R )-BMO condition (8).
0
2.3. Statement of the results. Let 1 p < . A nonnegative locally integrable function w belongs
≤ ∞
to the Muckenhoupt class A , say w A , if there exists a positive constant C so that
p p
∈
p−1
[w] := w(x)dx w−1/(p−1)(x)dx C, if1<p< ,
Ap ≤ ∞
B Q
(cid:16) (cid:17)(cid:16) (cid:17)
and
w(x)dx Cess-infw(x), if p=1,
B ≤ x∈B
for all balls B in Rn. We say that w A if w A for some p [1, ). We shall denote w(E) :=
∞ p
w(x)dx for any measurable set E ∈Rn. ∈ ∈ ∞
E ⊂
´ For a weight w and 0<q < we define
∞
1/q
Lqw(Ω)= f :kfkLqw(Ω) := ˆ |f(x)|qw(x)dx <∞ .
Ω
n (cid:16) (cid:17) o
We now record the following property of the Muckenhoupt weights in [22].
Lemma 2.5. Let w A . Then, there exist κ (0,1), and a constant c > 1 such that for any ball
∞ w w
∈ ∈
B and any measurable subset E B,
⊂
E κw
w(E) c | | w(B).
w
≤ B
(cid:16)| |(cid:17)
We now consider the continuous exponent function q() : Ω (0, ) satisfying the log-Ho¨lder conti-
· → ∞
nuity condition:
(11) q(x) q(y) ν(x y ), x,y Ω,
| − |≤ | − | ∀ ∈
where ν :[0, ) [0, ) is a non-decreasing function satisfying
∞ → ∞
1
(12) lim ν(r)log =0.
r→0+ r
(cid:16) (cid:17)
We also assume that there exist constants γ and γ such that
3 4
(13) 0<γ q(x) γ < , x Ω.
3 4
≤ ≤ ∞ ∀ ∈
Our first main result gives the weighted Lq(·) regularity for the solutions to problem (1).
Theorem 2.6. Let q() be defined as in (11), (12) and (13), w A and 0 < σ < min n(γ1−1),n .
· ∈ ∞ 0 n−1
Then there exists a positive constant δ = δ(n,Λ1,Λ2,p(),q(),w) such that the followingnholds. If thoe
· ·
domain Ω is a (δ,R ) Reifenberg flat domain with R > 0, and the nonlinearity a satisfies (2), (3) and
0 0
the small BMO condition (8), then for any weak solution u W1,p(·)(Ω) to the problem (1), the following
∈
estimate holds true
(14) ˆ |Du|q(x)w(x)dx≤C (|µ|(Ω)γ1σ−01 +|Ω|)n+1+ˆ |M1(µ)|p(qx()x−)1w(x)dx
Ω (cid:20) Ω (cid:21)
where C is a constant depending on n,Λ ,Λ ,p(),q(),R ,w,σ .
1 2 0 0
· ·
Or equivalently, we have
(15) ˆ |Du|(p(x)−1)q(x)w(x)dx. (|µ|(Ω)γ1σ−01 +|Ω|)n+1+ˆ |M1(µ)|q(x)w(x)dx .
Ω (cid:20) Ω (cid:21)
6 THEANHBUIANDXUANTHINHDUONG
Remark 2.7. In the particular case when q(x) q (0, ), the term (µ(Ω)γ1σ−01 + µ(Ω))n+1 in (14)
≡ ∈ ∞ | | | |
can be removed. More precisely, in this case we have
ˆ |Du|qw(x)dx .ˆ |M1(µ)|p(xq)−1w(x)dx.
Ω Ω
The proof can be done in the same manner as that of Theorem 2.6. However, we do not pursue it and
we would leave it to the interested reader.
We now have the following consequences of Theorem 2.6.
Corollary 2.8. Let q() be defined as in (11), (12) and (13) with 1 < γ γ < n. Then there exists
3 4
· ≤
a positive constant δ = δ(n,Λ ,Λ ,p(),q()) such that the following holds. If the domain Ω is a (δ,R )
1 2 0
· ·
Reifenbergflatdomain withR >0,andthenonlinearityasatisfies (2),(3)andthesmallBMOcondition
0
(8), then for any weak solution u W1,p(·)(Ω) to the problem (1) we obtain that
∈ 0
(16) dµ=fdx,f Lq(·)(Ω) Dup(·)−1 Lnn−qq(x(x))(Ω).
∈ ⇒| | ∈
In particular case when q(x) is independent of x, Theorem 2.6 deduces the following result.
Corollary 2.9. Let q (0, ), w A and 0 < σ < min n(γ1−1),n . Then there exists a positive
∈ ∞ ∈ ∞ 0 n−1
constant δ =δ(n,Λ1,Λ2,p(),q,w) such that the following holdns. If the doomain Ω is a (δ,R0) Reifenberg
·
flat domain with R >0, and the nonlinearity a satisfies (2), (3) and the small BMO condition (8), then
0
for any weak solution u W1,p(·)(Ω) to the problem (1) the following estimate holds true
∈ 0
(17) Dup(·)−1 C (µ(Ω)γ1σ−01 + Ω)n+q1 + M1(µ)
(cid:13)| | |(cid:13)Lqw(Ω) ≤ (cid:20) | | | | (cid:13) (cid:13)Lqw(Ω)(cid:21)
where C is a consta(cid:13)nt depending(cid:13)on n,Λ ,Λ ,p(),q,R ,w,σ . (cid:13) (cid:13)
(cid:13) (cid:13) 1 2 · 0 0 (cid:13) (cid:13)
As a consequence, for 1r − q1 = n1 and wq ∈A1+q/r′ if dµ=fdx,f ∈Lrwr(Ω), then we have
(18) Dup(·)−1 C ( f γ1σ−01 + Ω)n+q1 + f .
(cid:13)| | |(cid:13)Lqwq(Ω) ≤ (cid:20) k kL1(Ω) | | (cid:13) (cid:13)Lrwr(Ω)(cid:21)
Wenotethatthees(cid:13)(cid:13)timate(17)(cid:13)(cid:13)notonlygivestheLq-weightedestim(cid:13)(cid:13)ate(cid:13)(cid:13)for Dup(x)−1 butalsoimplies
| |
the estimate on Morrey space for Dup(x)−1. We now recall the definition of Morrey space.
| |
For 0 < q < and 0 < λ < n, the Morrey function spaces Lq;λ(Ω) is defined as the set of all
∞
measurable functions f such that
1
f = sup sup f < .
k kLq;λ(Ω) rλ/qk kLq(Br(x)∩Ω) ∞
x∈Ω0<r≤diamΩ
Usingastandardargument,seeforexample[38],fromtheweightedestimate(17)weobtainthefollowing
Morrey space estimate.
Corollary 2.10. Let q (0, ), λ (0,n) and 0<σ <min n(γ1−1),n . Then there exists a positive
∈ ∞ ∈ 0 n−1
constant δ =δ(n,Λ1,Λ2,p(),q,λ) such that the following holdsn. If the doomain Ω is a (δ,R0) Reifenberg
·
flat domain with R >0, and the nonlinearity a satisfies (2), (3) and the small BMO condition (8), then
0
for any weak solution u W1,p(·)(Ω) to the problem (1), the following estimate holds true
∈ 0
(19) Dup(·)−1 C (µ(Ω)γ1σ−01 + Ω)n+q1 + M1(µ)
| | | Lq;λ(Ω) ≤ | | | | Lq;λ(Ω)
(cid:13) (cid:13) (cid:20) (cid:13) (cid:13) (cid:21)
where C is a const(cid:13)ant dependin(cid:13)g on n,Λ ,Λ ,p(),q,R ,σ . (cid:13) (cid:13)
(cid:13) (cid:13) 1 2 · 0 0 (cid:13) (cid:13)
In general, if the measure µ is merely a Radon measure with finite total mass, the weak solution
u W1,p(·)(Ω) to (1) may not exist. In this situation, we employ the notion of SOLAs (Solution
∈ 0
Obtained as Limit of Approximations). It is well-known that these solution may not be in W1,p(·)(Ω),
0
but in W1,p(·)−1(Ω). In the particular case if µ W−1,p(·)(Ω), the dual space of W1,p(·)(Ω), it is well-
0 ∈ 0
known that there exists a unique solution u W1,p(·)(Ω) to (1), and in this case the SOLA and the
∈ 0
weak solution to (1) coincide. See for example [7]. From the above results, by a standardapproximation
procedure as in [7] we are able to obtain
Theorem 2.11. Let u W1,p(·)−1(Ω) be a SOLA to (1). Assume that all assumptions in the respective
∈ 0
statements hold true. Then Theorem 2.6 and Corollaries 2.8, 2.9 and 2.10 hold true.
REGULARITY ESTIMATES FOR ELLIPTIC EQUATIONS WITH NON-STANDARD GROWTH 7
3. Approximation results
Inthissection,wealwaysassumethatthenonlinearitysatisfies(2),(3),thesmallBMOnormcondition
(8) and the domain Ω is a (δ,R ) Reifenberg flat domain.
0
Let u be a weak solution to the problem (1). We now fix 0 < σ < min n(γ1−1),n . Then by a
0 n−1
standard argument as in the proof of [8, Theorem 1] there exists C =K(n,σ0,nΛ1,γ2,Ω)>o1 so that
Duσ0dx C(µ(Ω)γ1σ−01 + Ω).
ˆ | | ≤ | | | |
Ω
Hence, for any 0<q σ we have
0
≤
(20) ˆ |Du|qdx≤ˆ |Du|σ0 +1dx≤C(|µ|(Ω)γ1σ−01 +|Ω|)=:K0.
Ω Ω
For each r >0 and x Ω, we denote Ω :=Ω (x) and set
r r
∈
µ(Ω ) 1 µ(Ω ) 2−p+
F(µ,u,Ωr)=h|r|n−1r ip+Ωr−1 +h|r|n−1r i(cid:16) Ωr(|Du|+1)dx(cid:17) Ωrχ{p+Ωr≤2}+1,
where p+ =sup p(y) and
Ωr y∈Ωr
1, p+ 2,
χ{p+Ωr≤2} =(0, pΩ+Ωrr ≤>2.
3.1. Interior Estimates. Let x Ω and 0<R< R0∧Rω∧K0−1 so that B B (x ) Ω, where R
0 ∈ 10 2R ≡ 2R 0 ⊂ ω
is a constant in (7), and a b=min a,b . We set
∧ { }
p = inf p(x), p = sup p(x).
1 2
x∈B2R x∈B2R
Let u C1(Ω) be a solution to (1). We now consider the following equation
∈
diva(Dw,x)=0 in B ,
2R
(21)
w=u on ∂B ,
( 2R
We have the following estimate.
Proposition 3.1. Let w be a weak solution to (21). Then there exists a constant C so that
(22) D(u w)dx CF(µ,u,B ).
| − | ≤ 2R
B2R
As a consequence, we have
(23) Dwdx C Dudx+F(µ,u,B ) .
| | ≤ | | 2R
B2R (cid:20) B2R (cid:21)
Proof. We consider two cases: p 2 and 2 1 <p <2.
1 ≥ − n 1
Case 1: p 2. In this case, the inequality (22) was proved in [7, pp. 651–652].
1
≥
Case 2: p <2. It was proved in [4, Lemma 5.1] that
1
D(u w)dx. |µ|(B2R) p01−1 + |µ|(B2R) (Du +1)dx 2−p0 +1,
| − | Rn−1 Rn−1 | |
B2R h i h i(cid:16) B2R (cid:17)
where p =p(x ).
0 0
Note that µ(B ) µ(Ω) K R−1. Hence,
2R 0
| | ≤| | ≤ ≤
|µ|(B2R) p01−1 = |µ|(B2R) p21−1 |µ|(B2R) (p0−p21)−(pp20−1)
Rn−1 Rn−1 Rn−1
h i h|µ|(B2R)ip21−1hR−(p0n−(p12)−(pp2i0−)1)
≤ Rn−1
h|µ|(B2R)ip21−1R−(nγω1−(21R))2
≤ Rn−1
hC |µ|(B2Ri) p21−1,
≤ Rn−1
h i
noting that we used (7) in the last inequality.
8 THEANHBUIANDXUANTHINHDUONG
On the other hand, by (20) and (7),
µ(B2R) 2−p0 µ(B2R) 2−p2 p2−p0
| | (Du +1)dx = | | (Du +1)dx (Du +1)dx
Rn−1 | | Rn−1 | | | |
h i(cid:16) B2R (cid:17) h i(cid:16) B2R (cid:17) (cid:16) B2R (cid:17)
|µ|(B2R) (Du +1)dx 2−p2(R−nK )p2−p0
≤ Rn−1 | | 0
h i(cid:16) B2R (cid:17)
|µ|(B2R) (Du +1)dx 2−p2R−(n+1)(p2−p0)
≤ Rn−1 | |
h i(cid:16) B2R (cid:17)
|µ|(B2R) (Du +1)dx 2−p2R−(n+1)ω(2R)
≤ Rn−1 | |
h i(cid:16) B2R (cid:17)
µ(B2R) 2−p2
| | (Du +1)dx .
≤ Rn−1 | |
h i(cid:16) B2R (cid:17)
This completes our proof. (cid:3)
We now record the higher integrability result in [3, Theorem 5].
Lemma 3.2. Let w W1,p(·)(B ) be a weak solution to (21). Then there exists a constant σ∗ =
2R
∈
σ∗(n,Λ ,Λ ,γ , µ(Ω),Ω) so that for 0 σ <σ∗ and any q (0,1] there exists C >0 such that
1 2 2
| | ≤ ∈
1+σ 1/q
(24) Dwp(x)(1+σ)dx C Dwqp(x)dx +1 .
| | ≤ | |
(cid:16) BR (cid:17) (cid:20)(cid:16) B2R (cid:17) (cid:21)
Proof. It was proved in [3, Theorem 5] that for any σ (0,σ ) we have
1
∈
1+σ
Dwp(x)(1+σ)dx C Dwp(x)dx+1 ,
| | ≤ | |
(cid:16) BR (cid:17) (cid:20) B2R (cid:21)
with
−4(p2−p1)
σ =min 1,c Dwp(x)dx+1 p1 ,
1 ( 1(cid:16)ˆB2R| | (cid:17) )
for some constant c >0.
1
Notethatwecangetridofthedependenceoftheconstantσ on Dwp(x)dx. Todothiswerecall
1 B2R| |
an estimate in [7, p. 654] ´
γ
Dwp(x)dx CR−1−αβ Dwdx+1 1−β,
ˆ | | ≤ ˆ | |
BR (cid:16) B2R (cid:17)
where
κp p 1 κ p κ 1 n+1
α=n 2 1 ,β = 2 − p1,γ = 2 − ,κ= .
p − p × 1 1 p × 1 1 n
(cid:16) 1 (cid:17) 1 − p1 1 − p1 r
Hence,
Dwp(x)dx+1 4(p2p−1p1) CR−1−αβ×4(p2p−1p1) Dwdx+1 1−γβ×4(p2p−1p1).
ˆ | | ≤ ˆ | |
(cid:16) B2R (cid:17) (cid:16) B2R (cid:17)
Moreover,observe that
κγ γ 1 κ γ κ 1
α n 2 1 ,β 2 − γ2,γ 2 − .
≤ γ − ≤ γ × 1 1 ≤ γ × 1 1
(cid:16) 1 (cid:17) 1 − γ2 1 − γ2
As a consequence,
Dwp(x)dx+1 4(p2p−1p1) CR−c1(p2−p1) Dwdx+1 c2,
ˆ | | ≤ ˆ | |
(cid:16) B2R (cid:17) (cid:16) B2R (cid:17)
where c ,c are two constants independent of Dw and R.
1 2
Note that from (6) we have
R−c1(p2−p1) CR−c1γ(4R) C.
≤ ≤
On the other hand, from (20) and (23) by a simple manipulation we get
Dwdx+1 c(Ω, µ(Ω)).
ˆ | | ≤ | |
B2R
REGULARITY ESTIMATES FOR ELLIPTIC EQUATIONS WITH NON-STANDARD GROWTH 9
Therefore, there exists σ∗ =σ∗(n,Λ ,Λ ,γ , µ(Ω),Ω) so that
1 2 2
| |
σ >σ∗.
1
Hence, the desired estimate follows from Gehrings lemma in [26, Theorem 6.7]. (cid:3)
Consider the nonlinearity b(, ) associated to a(, ) defined by
· · · ·
(s2+ ξ 2)p2−2p(x)a(ξ,x),x B2R B2R(x0),
(25) b(ξ,x)= | | ∈ ≡
((s2+ ξ 2)p2−p2(x0)a(ξ,x0),x Rn B2R(x0).
| | ∈ \
Lemma 3.3. There exists R > 0 so that for any 0 < R < R , the nonlinearity b defined as above
a a
satisfies the following conditions:
(26) (s2+ ξ 2)1/2 Dξb(ξ,x) + b(ξ,x) 3Λ1(s2+ ξ 2)p22−1,
| | | | | |≤ | |
and
(27) Dξb(ξ,x)η,η Λ2(s2+ ξ 2)p22−2 ξ 2,
h i≥ 2 | | | |
for all x,ξ,η Rn.
∈
Proof. The proofof this lemma is similar to that in [13, p. 13]. For the convenience of reader,we sketch
it here.
From (2), by a simple calculation we have
(s2+ ξ 2)1/2 Dξb(ξ,x) + b(ξ,x) Λ1(1+γ(4r))(s2+ ξ 2)p22−1
| | | | | |≤ | |
:=Λ3(s2+ ξ 2)p22−1.
| |
It suffices to verify (27). We need to prove that there exist R > 0 so that for any 0 < R < R , the
a a
estimate (27) holds true for all x,ξ,η Rn.
∈
Indeed, if x B , we have
2R
∈
Dξb(ξ,x)η,η =(s2+ ξ 2)p2−2p(x) Dξa(ξ,x)η,η +(p2 p(x))ξ (s2+ ξ 2)p2−2p(x)−1 Dξa(ξ,x)η,η
h i | | h i − | | | | h i
:=I +I .
1 2
By (3), one gets that
I1 Λ2(s2+ ξ 2)p22−2 η 2.
≥ | | | |
Applying (2), we have
I2 Λ1(p2 p1)(s2+ ξ 2)p2−2p(x)−1(s2+ ξ 2)p(x2)−1 η 2
≥− − | | | | | |
ω(4R)Λ1(s2+ ξ 2)p22−2 η 2.
≥ | | | |
Hence,
Dξb(ξ,x)η,η [Λ2 ω(4R)Λ1](s2+ ξ 2)p22−2 η 2.
h i≥ − | | | |
Then the constant R can be chosen as a number satisfying γ(4R )< Λ2 .
a a 2Λ1
The case x Bc can be argued similarly. Hence, we complete the proof. (cid:3)
∈ 2R
We now consider the following equation
divb(Dh,x)=0 in B ,
R
(28)
h=w on ∂B ,
( R
where w is a weak solution to the problem (21).
Proposition 3.4. For any ǫ > 0 there exists R depending on ǫ only so that if h is a weak solution to
ǫ
(28) with 0<R< Rǫ∧R0∧Rω∧Ra∧K0−1, then we have
10
(29) D(h w)p2dx 1/p2 ǫ F(µ,u,B )+ Dudx .
| − | ≤ 2R | |
(cid:16) BR (cid:17) h B2R i
10 THEANHBUIANDXUANTHINHDUONG
Proof. We consider two cases: p <2 and p 2.
2 2
≥
Case 1: p <2. We first write
2
D(h w)p2 =(s2+ Dh2+ Dw2)−p2(p42−2)(s2+ Dh2+ Dw2)p2(p42−2) D(h w)p2.
| − | | | | | | | | | | − |
For τ >0, using Young’s inequality we obtain
1
D(h w)p2dx
| − |
BR
≤τ1 (s2+|Dh|2+|Dw|2)p22dx+c(τ1) (s2+|Dh|2+|Dw|2)p22−2|D(h−w)|2dx
(30) BR BR
cτ D(h w)p2dxdt+ Dwp2dx+1
≤ 1 | − | | |
h BR BR i
+c(τ1) (s2+|Dh|2+|Dw|2)p(x2)−2|D(h−w)|2dx.
BR
Note that, by (27), we have
(31) (s2+ Dh2+ Dw2)p22−2 D(h w)2dx C b(Dh,x) b(Dw,x),Dh Dw dx.
| | | | | − | ≤ h − − i
BR BR
Substituting (31) into (30), we obtain
D(h w)p2dx cτ D(h w)p2dx+cτ Dwp2dx+1
| − | ≤ 1 | − | 1 | |
(32) BR BR (cid:16) BR (cid:17)
+c(τ ) b(Dh,x) b(Dw,x),Dh Dw dx.
1 h − − i
BR
Moreover,from the definition of b(x,ξ) and (2) we have
b(Dh,x) b(Dw,x),D(h w) dx= a(Dw,x) b(Dw,x),D(h w) dx
h − − i h − − i
BR Br
= (s2+ Dw2)p2−2p(x) 1 a(Dw,x),D(h w) dx
| | − h − i
Br(cid:12) (cid:12)
C (cid:12)(cid:12) 1 (s2+ Dw2)p2−2p((cid:12)(cid:12)x) (s+ Dw)p(x)−1 D(h w)dx.
≤ − | | | | | − |
Brh i
Using the Mean Value Theorem, we obtain
b(Dh,x) b(Dw,x),D(h w) dx
h − − i
BR
≤C sup (p2−p(x))log(s2+|Dw|2)(s2+|Dw|2)θ(p2−2p(x))(s2+|Dw|)p(x)−1|D(h−w)|dx
BRθ∈[0,1]
Cω(4R) log(1+ Dw)(1+ Dw)p2−p(x)(1+ Dw)p(x)−1 D(h w)dx,
≤ | | | | | | | − |
BR
where in the last inequality we used the log-Ho¨lder condition (5).
Hence,
b(Dh,x) b(Dw,x),D(h w) dx Cω(4R) log(1+ Dw)(1+ Dw)p2−1 D(h w)dx.
h − − i ≤ | | | | | − |
BR BR
Using Young’s inequality, for τ >0, which will be fixed later, we obtain
2
b(Dh,x) b(Dw,x),D(h w) dx
h − − i
(33) BR
≤τ2ω(4R) |D(h−w)|p2dx+c(τ2)ω(4R) log(1+|Dw|)p2p−21(1+|Dw|)p2dx.
BR BR
Wenowapplytheinequalitylog(1+t)p2p−21 cα−p2p−21(1+t)α/4 fort>0andα (0,1)witht= Dw
≤ ∈ | |
and α=p ω(4R)/4 to conclude that
2
log(1+ Dw)p2p−21 c(p2ω(4R))−p2p−21(1+ Dw)p2ω(4R).
| | ≤ | |