Strong Unique Continuation for Solutions of a p x -Laplacian Problem

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International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 108671,16pages

doi:10.1155/2012/108671

Research Article

Strong Unique Continuation for Solutions of a p x -Laplacian Problem

Johnny Cuadro and Gabriel L ´opez

Mathematics Department, Universidad Aut´onoma Metropolitana, Avenue San Rafael Atlixco No. 186, Col. Vicentina Del. Iztapalapa, 09340 M´exico City, DF, Mexico

Correspondence should be addressed to Gabriel L ´opez,[email protected] Received 29 June 2012; Accepted 29 September 2012

Academic Editor: Chun-Lei Tang

Copyrightq2012 J. Cuadro and G. L ´opez. 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 Vx|u|^px−2u 0 inΩwhereVx ∈ L^N/pxΩ,Ωis a smooth bounded domain inR^N, and 1< px< NforxinΩ.

1. Introduction and Preliminary Results

LetΩbe an open, connected subset inR^N. Consider the Schr ¨odinger OperatorH−Δ V. IfHu0, and ifuvanishes of infinite order at one pointx0∈Ω see definitions inSection 3 imply thatu ≡ 0 inΩ, then Hhas the Strong Unique Continuation Property S.U.C.P. If, on the other hand,Hu 0 inΩ, andu 0 inΩ, an open subset ofΩ, imply thatu ≡0 in Ω, we say thatHhas the Weak Unique Continuation PropertyW.U.C.P. In 1939 Carleman 1 showed that H −Δ V has the S.U.C.P whenever V ∈ L^∞_locR². In order to prove 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 Kenig2 showed that ifn >2, pN/2 andV ∈L^p_loc, thenHhas the S.U.C.P.; Fabes et al. in3 gave a positive answer for a radial potentialVto Simon’s conjecture, which stated that for a potentialVin the Stummel-Kato class andu∈H¹ΩthenHhas the S.U.C.P. Other results were obtained by de Figueiredo and Gossez, but for Linear Elliptic Operators in the caseV ∈L^N/2Ω, N >2,4 . Also, Loulit extended this property toN 25 . More recently, Hadi and Tsouli6 proved Strong Unique Continuation Property for thep-Laplacian in the caseV ∈ L^N/pΩ, p < N andpconstant.

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 involvingpx-growth condition we refer to7–12 . The understanding of such physical models was facilitated by the development of variable Lebesgue and Sobolev spaces, L^px 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 L^px are special cases of Orlicz spacesL^ϕ originated by Nakano14 and developed by Musielak and Orlicz15,16 , wheref ∈ L^ϕ if and only if

ϕx,|fx|dx < ∞for a suitableϕ. For some interesting results on elliptic equation involving variable exponent growth conditions see 17–19 . We point out the presence of thepx-Laplace operator. This is a natural extension of thep-Laplace operator, withppositive constant. However, such generalizations are not trivial since thepx-Laplace operator possesses a more complicated structure thanp-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

Vx|u|^px−2u0 inΩ, 1.1

where 1 < px < N, V ∈ L^N/pxΩ and Ω ⊂ R^N is a bounded domain with smooth boundary.

Finally, we recall some definitions and basic properties of the variable exponent Lebesgue-Sobolev spacesL^p·ΩandW₀^1,p·Ω, whereΩis a bounded domain inR^N.

SetCΩ {h∈CΩ: min_x∈Ωhx>1}. For anyh∈CΩwe define

hsup

x∈Ωhx, h⁻inf

x∈Ωhx. 1.2

Forp∈CΩ, we introduce the variable exponent Lebesgue space:

L^p·Ω

u:uis a measurable real-valued function such that

Ω|ux|^pxdx <∞

, 1.3

endowed with the so-called Luxemburg norm:

|u|_p·inf

μ >0;

Ω

ux μ

^pxdx≤1

, 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 CΩ such that p1 ≤ p2 in Ω, then the embedding L^p²^·Ω → L^p¹^·Ω is continuous 20, Theorem 2.8 .

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LetL^p^·Ωbe the conjugate space ofL^p·Ω, obtained by conjugating the exponent pointwise, that is, 1/px 1/px 1, 20, Corollary 2.7 . For any u ∈ L^p·Ωand v ∈ L^p^·Ωthe following H ¨older type inequality

Ωuv dx ≤

1 p⁻ 1

p⁻

|u|_p·|v|_p· 1.5

is valid.

An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by thep·-modular of theL^p·Ωspace, which is the mappingρ_p· :L^p·Ω → Rdefined by

ρ_p·u

Ω|u|^pxdx. 1.6

Ifun, u∈L^p·Ωthen the following relations hold:

|u|_p·<1 1; >1⇐⇒ρ_p·u<1 1; >1, 1.7

|u|_p·>1⇒ |u|^p_p·⁻ ≤ρ_p·u≤ |u|^p_p· , 1.8

|u|_p·<1⇒ |u|^p_p· ≤ρ_p·u≤ |u|^p_p·⁻ , 1.9

|un−u|_p·−→0⇐⇒ρ_p·un−u−→0, 1.10

sincep < ∞. For a proof of these facts see20 . Spaces withp ∞have been studied by Edmunds et al.21 .

Next, we defineW₀^1,pxΩas the closure ofC₀^∞Ωunder the norm

u_px|∇u|_px. 1.11

The space W₀^1,pxΩ, · _px is a separable and reflexive Banach space. We note that if q ∈ CΩand qx < p^∗x for allx ∈ Ω then the embedding W₀^1,pxΩ → L^qxΩis continuous, wherep^∗x Npx/N−pxifpx < N orp^∗x ∞ifpx ≥ N20, Theorems 3.9 and 3.3 see also22, Theorems 1.3 and 1.1 .

The bounded variable exponent pis said to be Log-H ¨older continuous if there is a constantC >0 such that

px−p

y ≤ C

−log x−y 1.12

for allx, y∈R^N, such that|x−y| ≤1/2.

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A bounded exponentp is Log-H ¨older continuous in Ωif and only if there exists a constantC >0 such that

|B|^p^B⁻^−p^B ≤C 1.13

for every ballB⊂Ω 23, Lemma 4.1.6, page 101 .

As a result of the Log-H ¨older continuous condition we have

r^−p^B^−p⁻^B ≤C, C⁻¹r^−py≤r^px≤Cr^−py,

1.14

for allx, y ∈B:Bx0, r⊂Ωand the constantCdepends only on the constant Log-H ¨older continuous. It’s well known that Smooth Functions are dense in Variable Exponent Sobolev Spaces if the exponentpsatisfies the Log-H ¨older condition23, Proposition 11.2.3, page 346 .

2. On Fefferman’s Type Inequality

For everyu∈W₀^1,p·Ωthe norm Poincar´e inequality

|u|_Lp·Ω≤cdiamΩ|∇u|_Lp·, 2.1

c CN,Ω, clogp holdswe refer to 24 for notations and proofs. Nevertheless, the modular inequality

Ω|u|^pxdx≤C

Ω|∇u|^pxdx, ∀u∈W₀^1,p·Ω 2.2

not always holdssee18, Theorem 3.1 . It is known that2.2holds if, for instanceiN >1, and the functionft : pxotw is monotone 18, Theorem 3.4 with x_otw with an appropriate setting inΩ;iiif there exists a functionξ≥0 such that∇p· ∇ξ≥0,∇ξ/025, Theorem 1 ;iiiIf there existsa:Ω → R^Nbounded such that divax≥a0>0 for allx∈Ω andax· ∇px 0 for allx∈Ω,26, Theorem 1 . To the best of our knowledge necessary and suﬃcient conditions in order to ensure that

u∈W^1,p·infΩ/{0}

Ω|∇u|^px

Ω|u|^px >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 thatp·belongs to the Modular Poincar´e Inequality Class, MPICΩ, if there exist necessary conditions to ensure that

Ω|u|^px≤C

Ω|∇u|^px, ∀u∈W₀^1,p·Ω, 2.4

CCN,Ω, clogp>0 holds.

In27 Feﬀerman proved the following inequality:

R^N|ux|^p fx dx≤C

R^N|∇ux|^pdx ∀u∈C^∞₀ R^N

. 2.5

in the case p 2, assuming f in the Morrey’s space L^r,N−2rR^N, with 1 < r ≤ N/2.

Later in28 Schechter showed the same result takingf in the Stummel-Kato classSR^N. Chiarenza and Frasca29 generalized Fefferman’s result proving2.5under the assumption f ∈L^r,N−prR^N, with 1 < r < N/pand 1 < p < N. Zamboni in30 generalized Schecter’s result proving2.5under the assumptionf ∈M_pR^N, with 1< p < N. We stress out that it is not possible to compare the assumptionsf∈L^r,N−prR^N, the Morrey class, andf∈SR^N, the Stumel-Kato class. All the mentioned results were obtained for fixedp. 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. In31 Cuadro and L ópez proved inequality2.6for 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. Letpbe a Log-H¨older continuous exponent with 1 < px< N, andp∈MPICΩ.

LetV ∈ L¹_locΩwith 0< ε < Vxalmost everywhere. Then there exists a positive constantC CN,Ω, clogpsuch that

ΩVx|ux|^pxdx≤C

Ω|∇ux|^pxdx 2.6

for anyu∈W₀^1,pxΩ.

Proof. Letu∈W₀^1,pxΩsupported inBx0, r. Given thatV ∈L¹_locΩthe function

wx: x1

x⁰₁

Vξ1, x2, . . . , xndξ1, . . . , _x_N

x⁰_N

Vx1, . . . , x_N−1, ξNdξN

, 2.7

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where x0 x⁰₁, . . . , x⁰_N and x x1, . . . , xN ∈ Bx0, r, is well defined. Notice that _x_i

x⁰_iVx1, . . . , ξ_i, . . . , x_ndξi ∈Cx_i⁰, x_i fori1, . . . , NLemme VIII.232 so that divwx NVx. Moreover,

|Vx|_L¹_Bx₀_,r≥ _x₁

x⁰₁

· · · _x_N

x⁰_N

Vξdξn· · ·dξ₁, 2.8

whereξ ξ1, . . . , ξ_N. Therefore,|wx| ≤√

N|Vx|L¹Bx0,r. A direct calculation leads to

div

|u|^pxwx

|ux|^pxdivwx px|u|^px−2u∇u·wx |u|^pxlogu∇px·wx.

2.9

Now the Divergence Theorem implies

Bx0,r div|u|^pxwx 0, and so

Bx0,r|ux|^pxdivwxdx≤ p

Bx0,r|ux|^px−1|∇ux||wx|dx

Bx0,r|ux|^pxlog|ux| ∇px |wx|dx.

2.10

Set

I₁:p

Bx0,r|ux|^px−1|∇ux||wx|dx, I2:

Bx0,r|ux|^pxlog|ux| ∇px |wx|dx.

2.11

Now we estimateI2by distinguishing the case when|ux| ≤1 and|ux|>1. Notice that the relations

sup

0≤t≤1t^η logt <∞ 2.12

sup

t>1

t^−ηlogt <∞ 2.13

hold forη >0.

LetΩ1 :{x∈Br :|ux| ≤1}andΩ2 :{x∈Br :|ux|>1}, then for2.12and2.13 we have

I₂≤C₁

Ω1

|wx||ux|^px−η¹dxC₂

Ω2

|wx||ux|^pxη²dx. 2.14

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We can choosek∈Nsuch thatpx−1/k ≥p⁻. Sinceu∈L^p⁻Bx0, rand inΩ1,|ux| ≤1 we have

|ux|^px−1/n≤ |ux|^p⁻, 2.15

forn > k. The Lebesgue Dominated Convergence Theorem implies

nlim→ ∞

Ω1

|ux|^px−1/ndx

Ω1

|ux|^pxdx. 2.16

ForΩ2we can chooseksuch thatpx 1/k≤px^∗Npx/N−px. So

|ux|^px1/n≤ |ux|^px^∗, 2.17

n > k, andx∈Ω2. Sinceu∈L^px^∗Bx0, r 23, Theorem 8.3.1 we may use the Lebesgue Theorem again to obtain

nlim→ ∞

Ω2

|ux|^px1/ndx

Ω2

|ux|^pxdx. 2.18

Given thatp∈MPICΩwe have

I₂≤C

Bx0,r|u|^pxdx≤C

Bx0,r|∇u|^pxdx. 2.19

Now we estimateI₁by using the modular Young’s inequality24, equation3.2.21 :

I1≤pC1

Bx0,r|wx|^px/px−1|ux|^pxpC2

Bx0,r|∇ux|^px. 2.20 Again, sincep∈MPICΩwe obtain

I1≤C

Bx0,r|∇u|^pxdx. 2.21

Finally, recalling that divwx NVxwe get

N

Bxo,rVx|ux|^px≤C

Bx0,r|∇ux|^pxdx, 2.22

which leads to the claim of the theorem.

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3. Strong Unique Continuation

Consider the equation

Hu:div

|∇u|^px−2∇u

Vx|u|^px−2u0, x∈Ω, 3.1

u∈W_loc^1·pxΩ,1< px< N, V ∈L^N/pxΩ.

A weak solution of3.1is the functionu∈W_loc^1·pxΩsuch that

Ω|∇u|^px−2∇u· ∇ϕ dx

ΩVx|u|^px−2u·ϕ dx0, 3.2

for allϕ∈W₀^1,pxΩ.

The main interest of this section is to prove a unique continuation result for solutions of3.1according to the following definition.

Definition 3.1. A functionu∈L^px_loc Ωhas a zero of infinite order in thepx-mean at a point x₀∈Ωif, for eachk∈N,

|x−x0|R|u|^pxdxO R^k

. 3.3

Recall thatΩ⊂R^Nis a bounded open set. We want to prove estimates’ independency ofpfor bounded solutions. For this purpose we assume throughout this section that 1< p⁻ ≤ p < ∞andp is Lipschitz continuous. In particular,pis Log-H ¨older 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 involvingp, which would be impossible to control later, see24 .

Before proving Theorem 3.5 which is the main result of this paper we require the following Lemmas.

Lemma 3.2. Letpbe a Log-H¨older continuous exponent with 1 < px < N, andp ∈ MPICΩ.

LetV ∈L^N/px,0< < Vx,almost everywhere andu∈W₀^1,pxΩ. Then, for each_o>0, there existsKsuch that

ΩV|u|^pxdx≤o

Ω|∇u|^pxK

Ω|u|^pxdx. 3.4

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Proof. Leto>0 be given. We have

ΩVx|u|^px

{x:Vx>t}Vx|u|^px

{x:Vx≤t}Vx|u|^px

≤

{x:Vx>t}Vx|u|^pxt

Ω|u|^px

≤C

{x:Vx>t}|∇u|^pxt

Ω|u|^px,

3.5

where the last inequality follows from Theorem 2.2. Now, notice that the measure λE

E|∇u|^pxis absolutely continuous with respect to the Lebesgue measureμ. It follows that for 1 : o/C

Ω|∇u|^px >0 there existsδ >0 such that

E|∇u|^pxdμ < 1wheneverμE< δ.

Moreover, by Chebyshev’s type inequality,

μ{x:Vx> t}≤t^−N/p

ΩVx^N/px. 3.6 So takingtsuﬃciently big, we get the desired inequality.

Lemma 3.3. Letp:Ω → 1, Nbe an exponent with 1< p⁻≤p<∞and such thatp∈MPICΩ is Lipschitz continuous. Letube solution of3.1inΩ, andB_randB_2rtwo concentric balls contained inΩ. Then

Br

|∇u|^px≤ C r^px⁰

B2r

|u|^px, 3.7

where the constantCdoes not depend onr, x₀∈B_2randV ∈L^N/px.

Proof. Takeη ∈ C^∞₀ Ω, with suppη ⊂ B_2r,0 ≤ η ≤ 1 such thatηx 1 for anyx ∈ B_r and|∇η| ≤ C/r. We want to use as test functionψ η^pxu. To this end we show first that ψ ∈ W₀^1,pxΩ; it is clear that ψ ∈ L^pxΩsince uis solution of3.1. Furthermore, since 0η1 then|ηlogη|afor some constanta, so

∇ψ ≤ ∇uη^px upxη^px−1∇η uη^pxlogη∇px

≤ ∇uη^px upxη^px−1∇η uη^px−1∇pxηlogη

≤ ∇uη^px upxη^px−1∇η uη^px−1∇px a

≤ ∇uη^px uη^px−1 ∇η px a ∇px

≤ |∇u||u|

CpaL

≤ |∇u|Cp|u|.

3.8

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Hence,|∇ψ|^px2^p⁻¹|∇u|^pxC^p_p2^p⁻¹|u|^px. Therefore,|∇ψ| ∈L^pxΩ.

Now we can useψη^pxuas a test function to obtain 0

B2r

|∇u|^px−2∇u· ∇ψ dx

B2r

V|u|^px−2uψ dx

B2r

|∇u|^pxη^pxdx

B2r

|∇u|^px−2u∇u·

pxη^px−1∇ηη^pxlogη∇px dx

B2r

V η^px|u|^pxdx.

3.9

Let

I1:

B2r

|∇u|^px−2u∇u·

pxη^px−1∇ηη^pxlogη∇px dx,

I₂:

B2r

V η^px|u|^pxdx.

3.10

We can estimateI1by I1≤

B2r

|∇u|^px−2|u||∇u| pxη^px−1∇η η^pxlogη∇px dx

≤

B2r

|∇u| η ^px−1

|u| ∇η px a ∇px dx

≤

B2r

|∇u| η ^px−1

|u| ∇η paL dx

≤C_p

B2r

|∇u| η ^px−1

|u| ∇η dx

≤Cp

B2r

|∇u| η ^px 1

_px−1

|u| ∇η ^pxdx,

3.11

where the Young-type inequality

fg≤f^px/px−1 1

_px−1

g^px 3.12

was used in the last inequality. Moreover,

I1≤Cp

B2r

|∇u|^pxη^pxdxCp

B2r

1

_px−1

|u|^px ∇η ^pxdx

≤Cp

B2r

1

p−1

B2r

|u|^px ∇η ^pxdx.

3.13

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We estimateI2:

I2

B2r

V η^px|u|^pxdx

B2r

V ηu ^pxdx

≤

B2r

∇

uη ^pxdxC

B2r

ηu ^pxdx,

3.14

where Lemma3.2was used in the last inequality.

Now using the estimates forI₁andI₂, we have

B2r

|∇u|^pxη^pxdx≤Cp

B2r

1

p−1

B2r

|u|^px ∇η ^pxdx dx

B2r

∇

uη ^pxC

B2r

ηu ^pxdx

≤C1C₂

B2r

|∇u|^pxη^pxdx

^p

C₁C₂^p

B2r

|u|^px ∇η ^pxdxC

B2r

η^px|u|^pxdx 3.15

for 0< ≤1. By choosing≤min{1,1/2C1C2}, we have

B2r

|∇u|^pxη^pxdx≤ 1 ^p

B2r

|u|^px ∇η ^pxdx2C

B2r

|u|^pxη^pxdx

≤ 1 ^p

B2r

|u|^px C

r _px

dx2C

B2r

|u|^px C

r _px

dx.

3.16

Sincepxis Log-H ¨olderr^−px≤Cr^−px⁰for allx0∈B2r, then

B2r

|∇u|^pxη^pxdx≤ 1 ^p

B2r

|u|^px C

r _px₀

dx2C

B2r

|u|^px C

r _px₀

dx

≤ 1

^p C C^px⁰

r^px⁰

B2r

|u|^pxdx

≤ C

r^px⁰

B2r

|u|^pxdx.

3.17

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Therefore,

Br

|∇u|^pxdx≤ C r^px⁰

B2r

|u|^pxdx. 3.18

Lemma 3.4. Letu∈W^1,1Bx0, r, whereBx0, ris the ball of radiusr >0 inR^NandE{x∈ Bx0, r:ux 0}. Then there exists a constantβ >0 depending only onN, such that

D

|ux|dx≤βr^N

|E||D|^1/N

Bx0,r|∇ux|dx 3.19

for allBx0, r,uas above, and all mensurable setsD⊂Bx0, r.

Proof. See33, Lemma 3.4, page 54 .

Now we are ready to prove the main result in this paper.

Theorem 3.5. LetΩbe a bounded domain inR^N,p:Ω → 1, Nan exponent with 1< p⁻ ≤p <

∞and such thatp∈MPICΩis Lipschitz continuous, andu∈W_loc^1,pxΩa solution of 3.1. Ifu vanishes on setE⊂Ωof positive measure, thenuhas a zero of infinite order in thepx-mean.

Proof. We know that almost every point ofEis a point of density, letx₀be such a point, that is,

E^C∩Bx0, r

|Bx0, r| −→0, |E∩Bx0, r|

|Bx0, r| −→1 3.20

asr → 0.

LetB_r :Bx0, r. So for a given >0, there existsr₀r₀such that forr ≤r₀, E^C∩B_r

|Br| < , |E∩B_r|

|Br| >1−, 3.21

whereE^Cdenote the complement ofEinΩ. Takingr₀smaller if necessary, we may assume thatBr0⊂Ω. Sinceu0 onE, and usingLemma 3.4we have

Br

|ux|^pxdx

Br∩E^C|ux|^pxdx

Br∩E|ux|^pxdx

Br∩E^C|ux|^pxdx

≤β r^N

|Br∩E| Br∩E^C ^1/N

Br

∇

|ux|^px dx

≤Cβ r^N r^N−1

^1/N 1−

Br

∇

|ux|^px dx Cr^1/N

1−

Br

∇

|ux|^px dx.

3.22

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But|∇|ux|^px| ≤ |px|||ux|^px−1||∇ux||∇px|||ux|^pxlog|ux||. Hence,

Br

|ux|^pxdx≤Cr^1/N 1−

Br

p|ux|^px⁻¹|∇ux|dx

Br

L|ux|^px log|ux| dx

. 3.23

Let

I₁:

Br

|ux|^px−1|∇ux|dx, I₂:

Br

|ux|^pxlog|ux| dx.

3.24

I₁can be estimated using the Young type inequality with1/r:

Br

|ux|^px−1|∇ux|dx

Br

1

r|ux|^pxdx

Br

r^px−1|∇ux|^pxdx. 3.25

Now we estimateI₂ by distinguishing the case when|ux| ≤ 1 and |ux| > 1, using the relations2.2and2.13.

LetΩ1:{x∈Br :|ux| ≤1}andΩ2:{x∈Br :|ux|>1}, then I2≤C1

Ω1

|ux|^px−η¹dxC2

Ω2

|ux|^pxη²dx. 3.26

We can choosek ∈Nsuch thatpx−1/k ≥p⁻. Sinceu∈L^p⁻Bx0, rand inΩ1,|ux| ≤1 we have

|ux|^px−1/n≤ |ux|^p⁻, 3.27

forn > k. The Lebesgue Dominated Convergence Theorem implies

nlim→ ∞

Ω1

|ux|^px−1/ndx

Ω1

|ux|^pxdx. 3.28

ForΩ2we can chooseksuch thatpx 1/k≤px^∗Npx/N−px. So

|ux|^px1/n≤ |ux|^px^∗, 3.29

for n > k, and x ∈ Ω2. Since u ∈ L^px^∗Bx0, r 23, Theorem 8.3.1 we may use the Lebesgue Theorem again to obtain

nlim→ ∞

Ω2

|ux|^px1/ndx

Ω2

|ux|^pxdx. 3.30

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Therefore,

I₂ ≤C

Bx0,r|u|^pxdx. 3.31

Now, using estimates forI₁andI₂, and noticing that for 0< r <1 we haver^px< r^p⁻, we get

Br

|ux|^pxdx≤ ^1/N 1−

Cp

Br

|ux|^pxdx

Cpr^p⁻

Br

|∇ux|^pxdxC_p

Br

|ux|^pxdx

3.32

and by Lemma3.3we have

Br

|ux|^pxdx≤ ^1/N 1−

Cp

B2r

|ux|^pxdxCpr^p⁻r^−px⁰

B2r

|ux|^pxdx

Cp

B2r

|ux|^pxdx

≤C^1/N 1−

B2r

|ux|^pxdx,

3.33

whereCis independent ofεand ofr asr → 0. Note thatr^p⁻^−px⁰ < CwhereCis the Log- H ¨older 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 equation10to the end of the proof, or the proof of Theorem 2.16 , from inequality2.18to2.23, page 216; we include this last part of the proof for the sake of completeness. Setfr :

Br|ux|^px dx. Let us fixn ∈ Nand choose >0 such thatC^1/N/1− ≤2⁻ⁿ. Now, observe thatr₀depends onn, hence by the last inequality we deduce

fr≤2⁻ⁿf2r, forr≤r₀. 3.34

Iterating3.34, we get

f ρ

≤2^−knf 2^kρ

, if 2^k−1ρ≤r₀. 3.35

Thus, given that 0< r < r₀nand choosingk∈Nsuch that

2^−kr0≤r≤2^−k−1r0. 3.36

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From3.35, we conclude that

fr≤2^−knf 2^kr

≤2^−knf2r0, 3.37

and since 2^−k≤r/r₀, we get

fr≤ r

r0

_n

f2r0, 3.38

which shows thatx₀is a zero infinite order inpx-mean.

Acknowledgments

The authors want thank to Peter Häst ö 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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Strong Unique Continuation for Solutions of a p x -Laplacian Problem

Research Article

Strong Unique Continuation for Solutions of a p x -Laplacian Problem

Johnny Cuadro and Gabriel L ´opez

1. Introduction and Preliminary Results

2. On Fefferman’s Type Inequality

3. Strong Unique Continuation

Acknowledgments

References

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