Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 108671, 16 pages
doi:10.1155/2012/108671
Research Article
Strong Unique Continuation for Solutions of
a px-Laplacian Problem
Johnny Cuadro and Gabriel López
Mathematics Department, Universidad Autónoma Metropolitana, Avenue San Rafael Atlixco No. 186,
Col. Vicentina Del. Iztapalapa, 09340 México City, DF, Mexico
Correspondence should be addressed to Gabriel López, gabl@xanum.uam.mx
Received 29 June 2012; Accepted 29 September 2012
Academic Editor: Chun-Lei Tang
Copyright q 2012 J. Cuadro and G. López. 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.
We study the strong unique continuation property for solutions to the quasilinear elliptic equation
−div|∇u|px−2 ∇u V x|u|px−2 u 0 in Ω where V x ∈ LN/px Ω, Ω is a smooth bounded
domain in RN , and 1 < px < N for x in Ω.
1. Introduction and Preliminary Results
Let Ω be an open, connected subset in RN . Consider the Schrödinger Operator H −Δ V .
If Hu 0, and if u vanishes of infinite order at one point x0 ∈ Ω see definitions in Section 3
imply that u ≡ 0 in Ω, then H has the Strong Unique Continuation Property S.U.C.P. If,
on the other hand, Hu 0 in Ω, and u 0 in Ω , an open subset of Ω, imply that u ≡ 0 in
Ω, we say that H has the Weak Unique Continuation Property W.U.C.P. In 1939 Carleman
R2 . In order to prove
1 showed that H −Δ V has the S.U.C.P whenever V ∈ L∞
loc
this result he introduced a method, the so-called Carleman estimates, which has permeated
almost all the subsequent works in the subject. For instance, Jerison and Kenig 2 showed
p
that if n > 2, p N/2 and V ∈ Lloc , then H has the S.U.C.P.; Fabes et al. in 3 gave a positive
answer for a radial potential V to Simon’s conjecture, which stated that for a potential V in the
Stummel-Kato class and u ∈ H 1 Ω then H has the S.U.C.P. Other results were obtained by de
Figueiredo and Gossez, but for Linear Elliptic Operators in the case V ∈ LN/2 Ω, N > 2, 4.
Also, Loulit extended this property to N 2 5. More recently, Hadi and Tsouli 6 proved
Strong Unique Continuation Property for the p-Laplacian in the case V ∈ LN/p Ω, p < N
and p constant.
Equations involving variable exponent growth conditions have been intensively
discussed in the last decade. A strong motivation in the study of such kind of problems is
due to the fact that they can model with high accuracy various phenomena which arise from
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the study of elastic mechanics, electrorheological fluids, or image restoration; for information
on modeling physical phenomena by equations involving px-growth condition we refer
to 7–12. The understanding of such physical models was facilitated by the development
of variable Lebesgue and Sobolev spaces, Lpx and W 1,px , where px is a real-valued
function. Variable exponent Lebesgue spaces appeared for the first time in literature as early
as 1931 in an article by Orlicz 13. The spaces Lpx are special cases of Orlicz spaces Lϕ
originated by Nakano 14 and developed by Musielak and Orlicz 15, 16, where f ∈ Lϕ
if and only if ϕx, |fx|dx < ∞ for a suitable ϕ. For some interesting results on elliptic
equation involving variable exponent growth conditions see 17–19. We point out the
presence of the px-Laplace operator. This is a natural extension of the p-Laplace operator,
with p positive constant. However, such generalizations are not trivial since the px-Laplace
operator possesses a more complicated structure than p-Laplace operator; for example, it is
inhomogeneous.
In this paper we prove Strong Unique Continuation Property of the solutions of the
quasilinear elliptic equation:
−div |∇u|px−2 ∇u V x|u|px−2 u 0
in Ω,
1.1
where 1 < px < N, V ∈ LN/px Ω and Ω ⊂ RN is a bounded domain with smooth
boundary.
Finally, we recall some definitions and basic properties of the variable exponent
1,p·
Lebesgue-Sobolev spaces Lp· Ω and W0 Ω, where Ω is a bounded domain in RN .
Set C Ω {h ∈ CΩ : minx∈Ω hx > 1}. For any h ∈ C Ω we define
h sup hx,
h− inf hx.
1.2
x∈Ω
x∈Ω
For p ∈ C Ω, we introduce the variable exponent Lebesgue space:
Lp· Ω px
dx
<
∞
,
u : u is a measurable real-valued function such that
|ux|
Ω
1.3
endowed with the so-called Luxemburg norm:
|u|p·
ux px
inf μ > 0;
μ dx ≤ 1 ,
Ω
1.4
which is a separable and reflexive Banach space. For basic properties of the variable exponent
Lebesgue spaces we refer to 20. If 0 < |Ω| < ∞ and p1 , p2 are variable exponents in
C Ω such that p1 ≤ p2 in Ω, then the embedding Lp2 · Ω → Lp1 · Ω is continuous
20, Theorem 2.8.
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Let Lp · Ω be the conjugate space of Lp· Ω, obtained by conjugating the exponent
pointwise, that is, 1/px 1/p x 1, 20, Corollary 2.7. For any u ∈ Lp· Ω and v ∈
Lp · Ω the following Hölder type inequality
uv dx ≤ 1 1 |u| |v| p·
p ·
p− p −
Ω
1.5
is valid.
An important role in manipulating the generalized Lebesgue-Sobolev spaces is played
by the p·-modular of the Lp· Ω space, which is the mapping ρp· : Lp· Ω → R defined
by
ρp· u Ω
|u|px dx.
1.6
If un , u ∈ Lp· Ω then the following relations hold:
|u|p· < 1 1; > 1 ⇐⇒ ρp· u < 1 1; > 1,
1.7
p−
p
1.8
|u|p· < 1 ⇒ |u|p· ≤ ρp· u ≤ |u|p· ,
p
p−
1.9
|un − u|p· −→ 0 ⇐⇒ ρp· un − u −→ 0,
1.10
|u|p· > 1 ⇒ |u|p· ≤ ρp· u ≤ |u|p· ,
since p < ∞. For a proof of these facts see 20. Spaces with p ∞ have been studied by
Edmunds et al. 21.
1,px
Next, we define W0
Ω as the closure of C0∞ Ω under the norm
upx |∇u|px .
1,px
The space W0
1.11
Ω, · px is a separable and reflexive Banach space. We note that if
1,px
Ω → Lqx Ω is
q ∈ C Ω and qx < p∗ x for all x ∈ Ω then the embedding W0
∗
∗
continuous, where p x Npx/N − px if px < N or p x ∞ if px ≥ N 20,
Theorems 3.9 and 3.3 see also 22, Theorems 1.3 and 1.1.
The bounded variable exponent p is said to be Log-Hölder continuous if there is a
constant C > 0 such that
px − p y ≤
for all x, y ∈ RN , such that |x − y| ≤ 1/2.
C
− log x − y
1.12
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A bounded exponent p is Log-Hölder continuous in Ω if and only if there exists a
constant C > 0 such that
−
|B|pB −pB ≤ C
1.13
for every ball B ⊂ Ω 23, Lemma 4.1.6, page 101.
As a result of the Log-Hölder continuous condition we have
−
r −pB −pB ≤ C,
1.14
C−1 r −py ≤ r px ≤ Cr −py ,
for all x, y ∈ B : Bx0 , r ⊂ Ω and the constant C depends only on the constant Log-Hölder
continuous. It’s well known that Smooth Functions are dense in Variable Exponent Sobolev
Spaces if the exponent p satisfies the Log-Hölder condition 23, Proposition 11.2.3, page 346.
2. On Fefferman’s Type Inequality
1,p·
For every u ∈ W0
Ω the norm Poincaré inequality
|u|Lp· Ω ≤ c diamΩ|∇u|Lp· ,
2.1
c CN, Ω, c logp holds we refer to 24 for notations and proofs. Nevertheless, the
modular inequality
Ω
|u|px dx ≤ C
Ω
|∇u|px dx,
1,p·
∀u ∈ W0
Ω
2.2
not always holds see 18, Theorem 3.1. It is known that 2.2 holds if, for instance i N > 1,
and the function ft : pxo tw is monotone 18, Theorem 3.4 with xo tw with an
appropriate setting in Ω; ii if there exists a function ξ ≥ 0 such that ∇p · ∇ξ ≥ 0, ∇ξ / 0 25,
N
Theorem 1; iii If there exists a : Ω → R bounded such that div ax ≥ a0 > 0 for all x ∈ Ω
and ax · ∇px 0 for all x ∈ Ω, 26, Theorem 1. To the best of our knowledge necessary
and sufficient conditions in order to ensure that
inf
u∈W 1,p· Ω/{0}
Ω
|∇u|px
px
Ω |u|
>0
2.3
have not been obtained yet, except in the case N 1, 18, Theorem 3.2. The following
definition is in order.
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Definition 2.1. We say that p· belongs to the Modular Poincaré Inequality Class, MPICΩ,
if there exist necessary conditions to ensure that
px
Ω
|u|
≤C
Ω
1,p·
|∇u|px ,
∀u ∈ W0
Ω,
2.4
C CN, Ω, clog p > 0 holds.
In 27 Fefferman proved the following inequality:
|ux| fxdx ≤ C
p
RN
RN
|∇ux|p dx
∀u ∈ C0∞ RN .
2.5
in the case p 2, assuming f in the Morrey’s space Lr,N−2r RN , with 1 < r ≤ N/2.
Later in 28 Schechter showed the same result taking f in the Stummel-Kato class SRN .
Chiarenza and Frasca 29 generalized Fefferman’s result proving 2.5 under the assumption
f ∈ Lr,N−pr RN , with 1 < r < N/p and 1 < p < N. Zamboni in 30 generalized Schecter’s
p RN , with 1 < p < N. We stress out that it
result proving 2.5 under the assumption f ∈ M
is not possible to compare the assumptions f ∈ Lr,N−pr RN , the Morrey class, and f ∈ SRN ,
the Stumel-Kato class. All the mentioned results were obtained for fixed p. The theory for
a variable exponent spaces is a growing area but Modular Fefferman-type inequalities are
more scarce than Poincaré inequalities in variable exponent setting. In 31 Cuadro and López
proved inequality 2.6 for variable exponent spaces. We use such inequality in order to prove
S.U.C.P. We include the proof for the convenience of the reader.
Theorem 2.2. Let p be a Log-Hölder continuous exponent with 1 < px < N, and p ∈ MPICΩ.
Let V ∈ L1loc Ω with 0 < ε < V x almost everywhere. Then there exists a positive constant C CN, Ω, clog p such that
Ω
1,px
for any u ∈ W0
V x|ux|
px
dx ≤ C
Ω
|∇ux|px dx
2.6
Ω.
1,px
Proof. Let u ∈ W0
Ω supported in Bx0 , r. Given that V ∈ L1loc Ω the function
x
wx :
1
x10
V ξ1 , x2 , . . . , xn dξ1 , . . . ,
xN
0
xN
V x1 , . . . , xN−1 , ξN dξN ,
2.7
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International Journal of Mathematics and Mathematical Sciences
0
and x x1 , . . . , xN ∈ Bx0 , r, is well defined. Notice that
where
x0 x10 , . . . , xN
xi
0
0 V x1 , . . . , ξi , . . . , xn dξi ∈ Cxi , xi for i 1, . . . , N Lemme VIII.2 32 so that div wx xi
NV x. Moreover,
|V x|L1 Bx0 ,r ≥
where ξ ξ1 , . . . , ξN . Therefore, |wx| ≤
A direct calculation leads to
x1
x10
√
···
xN
0
xN
V ξdξn · · · dξ1 ,
2.8
N|V x|L1 Bx0 ,r .
div |u|px wx |ux|px div wx px|u|px−2 u∇u · wx
2.9
|u|px log u∇px · wx.
Now the Divergence Theorem implies
|ux|
px
div wxdx ≤ p
Bx0 ,r
Bx0 ,r
div|u|px wx 0, and so
|ux|px−1 |∇ux||wx|dx
Bx0 ,r
|ux|
px
log|ux|∇px|wx|dx.
2.10
Bx0 ,r
Set
I1 : p
|ux|px−1 |∇ux||wx|dx,
Bx0 ,r
|ux|
I2 :
px
log|ux|∇px|wx|dx.
2.11
Bx0 ,r
Now we estimate I2 by distinguishing the case when |ux| ≤ 1 and |ux| > 1. Notice
that the relations
sup tη log t < ∞
2.12
0≤t≤1
sup t−η log t < ∞
2.13
t>1
hold for η > 0.
Let Ω1 : {x ∈ Br : |ux| ≤ 1} and Ω2 : {x ∈ Br : |ux| > 1}, then for 2.12 and 2.13
we have
I2 ≤ C1
Ω1
|wx||ux|px−η1 dx C2
Ω2
|wx||ux|pxη2 dx.
2.14
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7
−
We can choose k ∈ N such that px − 1/k ≥ p− . Since u ∈ Lp Bx0 , r and in Ω1 , |ux| ≤ 1
we have
−
2.15
|ux|px−1/n ≤ |ux|p ,
for n > k. The Lebesgue Dominated Convergence Theorem implies
lim
n→∞
Ω1
|ux|px−1/n dx Ω1
|ux|px dx.
2.16
For Ω2 we can choose k such that px 1/k ≤ px∗ Npx/N − px. So
∗
|ux|px1/n ≤ |ux|px ,
2.17
∗
n > k , and x ∈ Ω2 . Since u ∈ Lpx Bx0 , r 23, Theorem 8.3.1 we may use the Lebesgue
Theorem again to obtain
lim
n→∞
Ω2
|ux|px1/n dx Ω2
|ux|px dx.
2.18
Given that p ∈ MPICΩ we have
|u|px dx ≤ C
I2 ≤ C
|∇u|px dx.
Bx0 ,r
2.19
Bx0 ,r
Now we estimate I1 by using the modular Young’s inequality 24, equation 3.2.21:
px/px−1
I1 ≤ p C1
|wx|
px
|ux|
|∇ux|px .
p C2
Bx0 ,r
Bx0 ,r
2.20
Again, since p ∈ MPICΩ we obtain
|∇u|px dx.
I1 ≤ C
Bx0 ,r
2.21
Finally, recalling that div wx NV x we get
V x|ux|px ≤ C
N
Bxo ,r
which leads to the claim of the theorem.
|∇ux|px dx,
Bx0 ,r
2.22
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International Journal of Mathematics and Mathematical Sciences
3. Strong Unique Continuation
Consider the equation
Hu : div |∇u|px−2 ∇u V x|u|px−2 u 0,
1·px
u ∈ Wloc
x ∈ Ω,
3.1
Ω, 1 < px < N, V ∈ LN/px Ω.
1·px
A weak solution of 3.1 is the function u ∈ Wloc
Ω such that
Ω
|∇u|px−2 ∇u · ∇ϕ dx Ω
V x|u|px−2 u · ϕ dx 0,
3.2
1,px
for all ϕ ∈ W0
Ω.
The main interest of this section is to prove a unique continuation result for solutions
of 3.1 according to the following definition.
px
Definition 3.1. A function u ∈ Lloc Ω has a zero of infinite order in the px-mean at a point
x0 ∈ Ω if, for each k ∈ N,
|x−x0 |R
|u|px dx O Rk .
3.3
Recall that Ω ⊂ RN is a bounded open set. We want to prove estimates’ independency
of p for bounded solutions. For this purpose we assume throughout this section that 1 < p− ≤
p < ∞ and p is Lipschitz continuous. In particular, p is Log-Hölder continuous. The new
feature in the estimate is the choice of a test function which includes the variable exponent.
This has both advantages and disadvantages: we need to assume that p is differentiable
almost everywhere, but, on the other hand, we avoid terms involving p , which would be
impossible to control later, see 24.
Before proving Theorem 3.5 which is the main result of this paper we require the
following Lemmas.
Lemma 3.2. Let p be a Log-Hölder continuous exponent with 1 < px < N, and p ∈ MPICΩ.
1,px
Ω. Then, for each o > 0, there
Let V ∈ LN/px , 0 < < V x, almost everywhere and u ∈ W0
exists K such that
Ω
V |u|
px
dx ≤ o
Ω
|∇u|
px
K
Ω
|u|px dx.
3.4
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Proof. Let o > 0 be given. We have
Ω
V x|u|
px
px
≤
{x:V x>t}
{x:V x>t}
V x|u|
V x|u|px t
≤C
{x:V x≤t}
V x|u|px
{x:V x>t}
Ω
|∇u|px t
Ω
|u|px
3.5
|u|px ,
the last inequality follows from Theorem 2.2. Now, notice that the measure λE where px
|∇u|
is absolutely
continuous with respect to the Lebesgue
measure μ. It follows that for
E
1 : o /C Ω |∇u|px > 0 there exists δ > 0 such that E |∇u|px dμ < 1 whenever μE < δ.
Moreover, by Chebyshev’s type inequality,
μ{x : V x > t} ≤ t−N/p
Ω
V xN/px .
3.6
So taking t sufficiently big, we get the desired inequality.
Lemma 3.3. Let p : Ω → 1, N be an exponent with 1 < p− ≤ p < ∞ and such that p ∈ MPICΩ
is Lipschitz continuous. Let u be solution of 3.1 in Ω, and Br and B2r two concentric balls contained
in Ω. Then
|∇u|px ≤
Br
C
r px0 |u|px ,
B2r
3.7
where the constant C does not depend on r, x0 ∈ B2r and V ∈ LN/px .
Proof. Take η ∈ C0∞ Ω, with sup pη ⊂ B2r , 0 ≤ η ≤ 1 such that ηx 1 for any x ∈ Br
and |∇η| ≤ C/r. We want to use as test function ψ ηpx u. To this end we show first that
1,px
Ω; it is clear that ψ ∈ Lpx Ω since u is solution of 3.1. Furthermore, since
ψ ∈ W0
0 η 1 then |η log η| a for some constant a, so
∇ψ ≤ ∇uηpx upxηpx−1 ∇η uηpx log η∇px
≤ ∇uηpx upxηpx−1 ∇η uηpx−1 ∇pxη log η
≤ ∇uηpx upxηpx−1 ∇η uηpx−1 ∇pxa
≤ ∇uηpx uηpx−1 ∇ηpx a∇px
≤ |∇u| |u| Cp aL
≤ |∇u| Cp |u|.
3.8
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p
Hence, |∇ψ|px 2p −1 |∇u|px Cp 2p −1 |u|px . Therefore, |∇ψ| ∈ Lpx Ω.
Now we can use ψ ηpx u as a test function to obtain
|∇u|px−2 ∇u · ∇ψ dx 0
B2r
|∇u|px−2 u∇u · pxηpx−1 ∇η ηpx log η∇px dx
|∇u|px ηpx dx B2r
V |u|px−2 uψ dx
B2r
B2r
3.9
V ηpx |u|px dx.
B2r
Let
|∇u|px−2 u∇u · pxηpx−1 ∇η ηpx log η∇px dx,
I1 :
B2r
3.10
I2 :
Vη
px
|u|
px
dx.
B2r
We can estimate I1 by
|∇u|px−2 |u||∇u| pxηpx−1 ∇η ηpx log η∇px dx
I1 ≤
B2r
px−1 |∇u|η
|u|∇η px a∇px dx
≤
B2r
px−1 |∇u|η
|u|∇η p aL dx
≤
B2r
3.11
px−1 |∇u|η
|u|∇η dx
≤ Cp
B2r
px
|∇u|η
≤ Cp
B2r
px−1
px
1
dx,
|u|∇η
where the Young-type inequality
fg ≤ f
px/px−1
px−1
1
g px
3.12
was used in the last inequality. Moreover,
px−1
px
1
|u|px ∇η dx
B2r
B2r
p −1 px
1
≤ Cp |∇u|px ηpx dx Cp
|u|px ∇η dx.
B2r
B2r
I1 ≤ Cp |∇u|px ηpx dx Cp
3.13
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We estimate I2 :
V ηpx |u|px dx
I2 B2r
px
V ηu dx
3.14
B2r
px
∇ uη dx C
≤
B2r
px
ηu dx,
B2r
where Lemma 3.2 was used in the last inequality.
Now using the estimates for I1 and I2 , we have
p −1 px
1
dx ≤ Cp |∇u| η dx Cp
|u|px ∇η dx dx
B2r
B2r
px
px
∇ uη ηu dx
C
|∇u|
px px
η
B2r
px px
B2r
B2r
≤ C1 C2 |∇u|
B2r
p C1 C2 p
px px
η
|u|
3.15
dx
px
∇η dx C
px B2r
ηpx |u|px dx
B2r
for 0 < ≤ 1. By choosing ≤ min{1, 1/2C1 C2 }, we have
|∇u|
B2r
px px
η
1
dx ≤ p
1
≤ p
|u|
B2r
px
∇η dx 2C
px |u|px ηpx dx
B2r
px
px
px C
px C
dx
2C
dx.
|u|
|u|
r
r
B2r
B2r
3.16
Since px is Log-Hölder r −px ≤ Cr −px0 for all x0 ∈ B2r , then
px0 px0 1
px C
px C
dx
2C
dx
|u|
|u|
r
r
p B2r
B2r
px0 1
C
≤
C
|u|px dx
p
r px0 B2r
C
≤ px |u|px dx.
r 0 B2r
|∇u|px ηpx dx ≤
B2r
3.17
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International Journal of Mathematics and Mathematical Sciences
Therefore,
|∇u|px dx ≤
Br
C
|u|px dx.
r px0 B2r
3.18
Lemma 3.4. Let u ∈ W 1,1 Bx0 , r, where Bx0 , r is the ball of radius r > 0 in RN and E {x ∈
Bx0 , r : ux 0}. Then there exists a constant β > 0 depending only on N, such that
|ux|dx ≤ β
D
rN
|D|1/N
|E|
|∇ux|dx
3.19
Bx0 ,r
for all Bx0 , r, u as above, and all mensurable sets D ⊂ Bx0 , r.
Proof. See 33, Lemma 3.4, page 54.
Now we are ready to prove the main result in this paper.
Theorem 3.5. Let Ω be a bounded domain in RN , p : Ω → 1, N an exponent with 1 < p− ≤ p <
1,px
∞ and such that p ∈ MPICΩ is Lipschitz continuous, and u ∈ Wloc Ω a solution of 3.1. If u
vanishes on set E ⊂ Ω of positive measure, then u has a zero of infinite order in the px-mean.
Proof. We know that almost every point of E is a point of density, let x0 be such a point, that
is,
C
E ∩ Bx0 , r
−→ 0,
|Bx0 , r|
|E ∩ Bx0 , r|
−→ 1
|Bx0 , r|
3.20
as r → 0.
Let Br : Bx0 , r. So for a given > 0, there exists r0 r0 such that for r ≤ r0 ,
C
E ∩ Br < ,
|Br |
|E ∩ Br |
> 1 − ,
|Br |
3.21
where EC denote the complement of E in Ω. Taking r0 smaller if necessary, we may assume
that Br0 ⊂ Ω. Since u 0 on E, and using Lemma 3.4 we have
|ux|px dx Br
Br
∩EC
Br ∩EC
|ux|px dx Br ∩E
|ux|px dx
|ux|px dx
1/N r N C
≤β
∇ |ux|px dx
Br ∩ E |Br ∩ E|
Br
N
1/N
r ≤ Cβ N−1
∇ |ux|px dx
1 − Br
r
1/N
Cr
∇ |ux|px dx.
1 − Br
3.22
International Journal of Mathematics and Mathematical Sciences
13
But |∇|ux|px | ≤ |px|||ux|px−1 ||∇ux| |∇px|||ux|px log |ux||. Hence,
|ux|
Br
px
1/N
dx ≤ Cr
1−
p |ux|
px −1
|∇ux|dx Br
L|ux|
log|ux| dx .
px Br
3.23
Let
|ux|px−1 |∇ux|dx,
I1 :
Br
I2 :
|ux|px log |ux|dx.
3.24
Br
I1 can be estimated using the Young type inequality with 1/r:
|ux|px−1 |∇ux|dx Br
Br
1
|ux|px dx r
r px−1 |∇ux|px dx.
Br
3.25
Now we estimate I2 by distinguishing the case when |ux| ≤ 1 and |ux| > 1, using the
relations 2.2 and 2.13.
Let Ω1 : {x ∈ Br : |ux| ≤ 1} and Ω2 : {x ∈ Br : |ux| > 1}, then
I2 ≤ C1
Ω1
|ux|px−η1 dx C2
Ω2
|ux|pxη2 dx.
3.26
−
We can choose k ∈ N such that px − 1/k ≥ p− . Since u ∈ Lp Bx0 , r and in Ω1 ,|ux| ≤ 1
we have
−
3.27
|ux|px−1/n ≤ |ux|p ,
for n > k. The Lebesgue Dominated Convergence Theorem implies
lim
n→∞
Ω1
|ux|px−1/n dx Ω1
|ux|px dx.
3.28
For Ω2 we can choose k such that px 1/k ≤ px∗ Npx/N − px. So
∗
|ux|px1/n ≤ |ux|px ,
3.29
∗
for n > k , and x ∈ Ω2 . Since u ∈ Lpx Bx0 , r 23, Theorem 8.3.1 we may use the
Lebesgue Theorem again to obtain
lim
n→∞
Ω2
|ux|px1/n dx Ω2
|ux|px dx.
3.30
14
International Journal of Mathematics and Mathematical Sciences
Therefore,
|u|px dx.
I2 ≤ C
3.31
Bx0 ,r
−
Now, using estimates for I1 and I2 , and noticing that for 0 < r < 1 we have r px < r p , we get
px
|ux|
Br
1/N
Cp
dx ≤
|ux|px dx
1−
Br
Cp r p
−
|∇ux|px dx Cp
Br
3.32
|ux|px dx
Br
and by Lemma 3.3 we have
px
|ux|
Br
1/N
px
p− −px0 Cp
dx ≤
|ux| dx Cp r r
|ux|px dx
1−
B2r
B2r
|ux|px dx
Cp
3.33
B2r
≤C
1/N
1−
|ux|px dx,
B2r
−
where C is independent of ε and of r as r → 0. Note that r p −px0 < C where C is the LogHölder constant. From this point the argument in the proof is standard, see, for instance, in
4 the proof of Lemma 1, page 344-345 from equation 10 to the end of the proof, or the
proof of Theorem 2.1 6, from inequality 2.18 to 2.23,
page 216; we include this last part
of the proof for the sake of completeness. Set fr : Br |ux|px dx. Let us fix n ∈ N and
choose > 0 such that C1/N /1 − ≤ 2−n . Now, observe that r0 depends on n, hence by
the last inequality we deduce
fr ≤ 2−n f2r,
for r ≤ r0 .
3.34
if 2k−1 ρ ≤ r0 .
3.35
Iterating 3.34, we get
f ρ ≤ 2−kn f 2k ρ ,
Thus, given that 0 < r < r0 n and choosing k ∈ N such that
2−k r0 ≤ r ≤ 2−k−1 r0 .
3.36
International Journal of Mathematics and Mathematical Sciences
15
From 3.35, we conclude that
fr ≤ 2−kn f 2k r ≤ 2−kn f2r0 ,
3.37
and since 2−k ≤ r/r0 , we get
fr ≤
r
r0
n
f2r0 ,
3.38
which shows that x0 is a zero infinite order in px-mean.
Acknowledgments
The authors want thank to Peter Hästö for his careful reading and corrections to a draft of
this paper. J. Cuadro was supported by CONACYT México’s Ph.D. Schoolarship.
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