STRONG UNIQUE CONTINUATION OF EIGENFUNCTIONS FOR p-LAPLACIAN OPERATOR

(1)

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 functions 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.

(2)

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 integral 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).

(3)

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.

(4)

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

[1] D. G. de Figueiredoand J.-P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Diﬀerential Equations 17 (1992), no. 1-2, 339–346.

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:hadi@sciences.univ-oujda.ac.ma

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

E-mail address:tsouli@sciences.univ-oujda.ac.ma

(5)

Special Issue on

Singular Boundary Value Problems for Ordinary Differential Equations

Call for Papers

The purpose of this special issue is to study singular boundary value problems arising in di

ﬀ

erential equations and dynamical systems. Survey articles dealing with interac- tions between diﬀerent fields, applications, and approaches of boundary value problems and singular problems are welcome.

This Special Issue will focus on any type of singularities that appear in the study of boundary value problems. It includes:

•

Theory and methods

•

Mathematical Models

•

Engineering applications

•

Biological applications

•

Medical Applications

•

Finance applications

•

Numerical and simulation applications

Before submission authors should carefully read over the journal’s Author Guidelines, which are located at http://www.hindawi.com/journals/bvp/guidelines.html. Au- thors should follow the Boundary Value Problems manu- script format described at the journal site http://www .hindawi.com/journals/bvp/. Articles published in this Spe- cial Issue shall be subject to a reduced Article Proc- essing Charge of C200 per article. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking Sys- tem at http://mts.hindawi.com/ according to the following timetable:

Manuscript Due May 1, 2009 First Round of Reviews August 1, 2009 Publication Date November 1, 2009

Lead Guest Editor

Juan J. Nieto,

Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de

Compostela, Santiago de Compostela 15782, Spain;

juanjose.nieto.roig@usc.es

Guest Editor

Donal O’Regan,

Department of Mathematics, National University of Ireland, Galway, Ireland;

donal.oregan@nuigalway.ie

Hindawi Publishing Corporation http://www.hindawi.com