STRONG UNIQUE CONTINUATION OF EIGENFUNCTIONS FOR p-LAPLACIAN OPERATOR

IJMMS 25:3 (2001) 213–216 PII. S0161171201004744 http://ijmms.hindawi.com

© Hindawi Publishing Corp.

STRONG UNIQUE CONTINUATION OF EIGENFUNCTIONS FOR p-LAPLACIAN OPERATOR

ISLAM EDDINE HADI and N. TSOULI (Received 28 January 2000)

Abstract.We show the strong unique continuation property of the eigenfunctions for p-Laplacian operator in the casep < N.

2000 Mathematics Subject Classification. Primary 35J15.

1. Introduction. This paper is primarily concerned with the problem:

−div

|∇u|^p−2∇u

+V|u|^p−2u=0 inΩ, (1.1)

whereΩis a bounded domain inR^Nand the weight functionsVis assumed tobe not equivalent tozeroand tolie inL^N/p(Ω).

Also, we know that the unique continuation property is defined by a different form.

In this work, we are interested to study a family of functions which enjoys the strong unique continuation property (SUCP), that is, functions besides possibly the zero func- tions has a zero of infinite order.

Definition1.1. A functionu∈L^p(Ω)has a zero of infinite order inp-mean at x0∈Ω, if for eachn∈N,

|x−x0|≤R|u|^p=0 Rⁿ

asR →0. (1.2)

There is an extensive literature on unique continuation. We refer to the work of Jerison-Kenig on the unique continuation for Shrödinger operators (cf. [3]). The same work is done by Gossez and Figueiredo, but for linear elliptic operator in the caseV∈ L^N/2, whereN >2, (cf. [1]). Also, Loulit extended this property toN=2 by introducing Orlicz’s space, (cf. [2, 5]). In this work, we generalize this property for thep-Laplacian in the caseV∈L^N/p(Ω)andp < N.

2. Strong unique continuation theorem. In this section, we proceed to establish the strong unique continuation property of the eigenfunctions for thep-Laplacian operator in the caseV∈L^N/p(Ω)andp < N.

Theorem 2.1. Letu∈W_loc^1,p(Ω) solution of (1.1). If u=0 on a setE of positive measure, thenuhas a zero of infinite order inp-mean.

214 I. E. HADI AND N. TSOULI To prove this theorem we need the following lemmas.

Lemma2.2. Letg∈W0^1,p(Ω)andV∈L^N/p. Then for each >0there exists a positive constantksuch that

ΩV|g|^p≤

Ω|∇g|^p+k

Ω|g|^p. (2.1)

Proof. Set

G= x∈ Ω

V(x)≥c

. (2.2)

So

ΩV|g|^p≤

GV|g|^p+k

Ω|g|^p. (2.3)

By using the Hölder and Poincaré’s inequalities, we get

ΩV|g|^p≤cχGV^N/p_L

Ω|∇g|^p+k

Ω|g|^p. (2.4)

But·is absolutely continuous. So, given >0, there existsksuch thatcχGV ≤. Which gives immediately the inequality (2.1).

Lemma2.3. LetBr andB2r be two concentric balls contained inΩ. Then

Br|∇u|^p≤ c r^p

B2r|u|^p, (2.5)

where the constantcdoes not depend onr.

Proof. Takeϕ∈C₀^∞(Ω), with suppϕ⊂B2r,ϕ(x)=1 fo rx∈Br and|∇ϕ| ≤c/r. Usingϕ^puas a test function in (1.1), we get

B2r−div

|∇u|^p−2∇u ϕ^pu+

B2rV|u|^p−2uϕ^pu=0. (2.6)

So

B2r

|∇u|^pϕ^p= −p

B2r

|∇u|^p−2ϕ^p−2∇u·∇ϕ(ϕu)−

B2r

V|ϕu|^p. (2.7) Using Young’s inequalities for(((p−1)/p)+1/p=1), we can estimate the first inte- gral in the right-hand side of (2.7) by

(p−1)^p/(p−1)

B2r|∇u|^pϕ^p+^−p

B2r|∇ϕ|^p|u|^p. (2.8) Also by the result of Lemma 2.2, we can estimate the second integral in the right-hand side of (2.7) by

B2r

∇(ϕu)^p+c

B2r|ϕu|^p. (2.9)

Using these estimates in (2.7), we have

B2r|∇u|^pϕ^p≤

(p−1)^p/(p−1)+

B2r|∇u|^p|ϕ|^p +

^−p+

B2r

|u|^p|∇ϕ|^p+c

B2r

|u|^p|ϕ|^p. (2.10) Using the fact that|∇ϕ| ≤c/r, |ϕ| ≤c/r, andϕ=1 inBr, we have immediately inequality (2.5).

STRONG UNIQUE CONTINUATION OF EIGENFUNCTIONS... 215 Lemma2.4. Letu∈W^1,1(Br), whereBr is the ball of radius r inR^N and letE= {x∈Br:u(x)=0}. Then there exists a constantβdependingonly onNsuch that

A|u| ≤βr^N

|E||A|^1/N

Br|∇u| (2.11)

for all ballBr,uas above and all measurable setsA⊂Br. Toprove this lemma see [4].

Proof of Theorem2.1. We know that almost every point ofEis a point of den- sity ofE. Letx0∈Ebe such a point. This means that

limr→0

E∩Br

Br =1, (2.12)

whereBr denotes the ball of radiusr centered atx0and|S|denotes the Lebesgue measure of a setS. So, given >0 there is anr0=r0()such that

E^c∩Br

Br < , E∩Br

Br >1− forr≤r0, (2.13) whereE^cdenotes the complement of the setE. Takingr0smaller, if necessary, we can assumeBr0⊂Ω. Sinceu=0 o nE, by Lemma 2.4 and (2.13) we have

Br|u|^p=

Br∩E^c|u|^p≤β r^N

E∩Br|E^c∩Br|^1/N

Br

∇(u)^p

≤pβ r^N

Br^(1−1/N)1/N 1−

Br|u|^p−1|∇u|.

(2.14)

By Hölder’s inequality

Br|u|^p≤c^1/N 1−r

Br|∇u|^{p 1/p}

Br|u|^{p (p−1)/p}, (2.15) and by using the Young’s inequality, we get

Br|u|^p≤c^1/N 1−r

r^p−1

Br|∇u|^p+p−1 r

Br|u|^p . (2.16) Finally, by Lemma 2.3, we have

Br|u|^p≤c^1/N 1−

B2r|u|^p, (2.17)

wherecis independent ofand ofrasr→0.

216 I. E. HADI AND N. TSOULI Now let us introduce the following functions:

f (r )=

Br|u|^p. (2.18)

And let us fix n∈N, choose >0 such that c^1/N

/(1−)≤2⁻ⁿ. Observe that consequentlyr0depends onn. Then (2.17) can be written as

f (r )≤2⁻ⁿf (2r ) forr≤r0. (2.19) Iterating (2.19), we get

f (ρ )≤2^−knf 2^kρ

, if 2^k−1ρ≤r0. (2.20) Now given 0< r < r0(n)and choosek∈Nsuch that

2^−kr0≤r≤2^−k+1r0. (2.21) From (2.20), we obtain

f (r )≤2^−knf 2^kr

≤2^−knf 2r0

. (2.22)

Since 2^−k≤r /r0, we finally obtain f (r )≤

r r0

nf 2r0

, (2.23)

which shows thatx0is a zeroinfinite order inp-mean.

References

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MR 93b:35098. Zbl 777.35042.

[2] J.-P. Gossez and A. Loulit,A note on two notions of unique continuation, Bull. Soc. Math.

Belg. Sér. B45(1993), no. 3, 257–268. MR 96k:35034. Zbl 828.35035.

[3] D. Jerison and C. E. Kenig,Unique continuation and absence of positive eigenvalues for Schrödinger operators.With an appendix by E. M. Stein, Ann. of Math. (2)121(1985), no. 3, 463–494. MR 87a:35058. Zbl 593.35119.

[4] O. A. Ladyzhenskaya and N. N. Ural’tseva,Linear and Quasilinear Elliptic Equations, Trans- lated from the Russian by Scripta Technica, Inc. Leon Ehrenpreis (trans. ed.), Aca- demic Press, New York, London, 1968. MR 39#5941. Zbl 164.13002.

[5] A. Loulit,Inégalités avec Poids et Problèmes de Continuation Unique, Thèse de Doctorat, Université Libre de Bruxelles, 1995 (French).

Islam Eddine Hadi: Département de Mathématiques et Informatique Faculté des Sci- ences, Université MohamedI, Oujda, Morocco

E-mail address:[email protected]

N. Tsouli: Département de Mathématiques et Informatique Faculté des Sciences, Université MohamedI, Oujda, Morocco

E-mail address:[email protected]