Semiclassical standing waves with clustering peaks for nonlinear Schrodinger equations

104 Pages·2014·0.821 MB·English

by  Jaeyoung Byeon, Kazunaga Tanaka

MEMOIRS
of the
American Mathematical Society
Volume 229 • Number 1076 (third of 5 numbers) • May 2014
Semiclassical Standing Waves with
Clustering Peaks for Nonlinear
Schro¨dinger Equations
Jaeyoung Byeon
Kazunaga Tanaka
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 229 • Number 1076 (third of 5 numbers) • May 2014
Semiclassical Standing Waves with
Clustering Peaks for Nonlinear
Schro¨dinger Equations
Jaeyoung Byeon
Kazunaga Tanaka
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Byeon,Jaeyoung,1966-
Semiclassical standing waves with clustering peaks for nonlinear Schr¨odinger equations /
JaeyoungByeon,KazunagaTanaka.
pages cm. – (Memoirs of the american mathematical society, ISSN 0065-9266 ; volume 229,
number1076)
“May2014,volume229,number1076(thirdof5numbers).”
Includesbibliographicalreferences.
ISBN978-0-8218-9163-6(alk. paper)
1.Gross-Pitaevskiiequations. 2.Schro¨dingerequation. 3.Standingwaves. 4.Clusteranal-
ysis. I.Tanaka,Kazunaga,1959- II.Title.
QC174.26.W28B94 2014
530.12(cid:2)4–dc23 2013051230
DOI:http://dx.doi.org/10.1090/memo/1076
Memoirs of the American Mathematical Society
Thisjournalisdevotedentirelytoresearchinpureandappliedmathematics.
Subscription information. Beginning with the January 2010 issue, Memoirs is accessible
from www.ams.org/journals. The 2014 subscription begins with volume 227 and consists of six
mailings,eachcontainingoneormorenumbers. Subscriptionpricesareasfollows: forpaperdeliv-
ery,US$827list,US$661.60institutionalmember;forelectronicdelivery,US$728list,US$582.40
institutionalmember. Uponrequest,subscriberstopaperdeliveryofthisjournalarealsoentitled
to receive electronic delivery. If ordering the paper version, add US$10 for delivery within the
United States; US$69 for outside the United States. Subscription renewals are subject to late
fees. Seewww.ams.org/help-faqformorejournalsubscriptioninformation. Eachnumbermaybe
orderedseparately;please specifynumber whenorderinganindividualnumber.
Back number information. Forbackissuesseewww.ams.org/bookstore.
Subscriptions and orders should be addressed to the American Mathematical Society, P.O.
Box 845904, Boston, MA 02284-5904USA. All orders must be accompanied by payment. Other
correspondenceshouldbeaddressedto201CharlesStreet,Providence,RI02904-2294USA.
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
[email protected].
MemoirsoftheAmericanMathematicalSociety (ISSN0065-9266(print);1947-6221(online))
ispublishedbimonthly(eachvolumeconsistingusuallyofmorethanonenumber)bytheAmerican
MathematicalSocietyat201CharlesStreet,Providence,RI02904-2294USA.Periodicalspostage
paid at Providence, RI.Postmaster: Send address changes to Memoirs, AmericanMathematical
Society,201CharlesStreet,Providence,RI02904-2294USA.
(cid:2)c 2013bytheAmericanMathematicalSociety. Allrightsreserved.
Copyrightofindividualarticlesmayreverttothepublicdomain28years
afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles.
(cid:2)
ThispublicationisindexedinMathematicalReviewsR,Zentralblatt MATH,ScienceCitation
Index(cid:2)R,ScienceCitation IndexTM-Expanded,ISI Alerting ServicesSM,SciSearch(cid:2)R,Research
(cid:2) (cid:2) (cid:2)
AlertR,CompuMathCitation IndexR,Current ContentsR/Physical, Chemical& Earth
Sciences. ThispublicationisarchivedinPortico andCLOCKSS.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 191817161514
Dedicated to Professor Hiroshi Matano
on the occasion of his 60th birthday

Contents
Chapter 1. Introduction and results 1
Chapter 2. Preliminaries 9
2.1. Notation 9
2.2. Limit problems 9
2.3. Estimate of L (u) with relation to the Pohozaev identity 10
m0
2.4. Functional setting 11
2.5. Choice of a neighborhood O of M 12
2.6. Control on small parts of u∈H1(RN) 12
Chapter 3. Local centers of mass 15
3.1. Local centers of mass 15
3.2. Some functions Λ(u), Ξ(u), Θ (u) in terms of local centers of mass 18
ε
Chapter 4. Neighborhood Ω (ρ,R,β) and minimization for a tail of u in Ω 21
ε ε
4.1. A choice of parameters and minimization 21
4.2. Invariant new neighborhoods 23
4.3. Width of a set Z(cid:2)(ρ(cid:3)(cid:3),R(cid:3)(cid:3))\Z(cid:2)(ρ(cid:3),R(cid:3)) 25
Chapter 5. A gradient estimate for the energy functional 29
5.1. ε-dependent concentration-compactness argument 29
5.2. A gradient estimate 33
5.3. Gradient flow of the energy functional Γ 37
ε
Chapter 6. Translation flow associated to a gradient flow of V(x) on RN 39
6.1. Apseudo-gradientflowonN3β0(O)(cid:4)0 associatedtoV(x1)+···+V(x(cid:4)0) 39
6.2. Definition of a translation operator 40
6.3. Properties of the translation operator 42
Chapter 7. Iteration procedure for the gradient flow and the translation flow 49
Chapter 8. An (N +1)(cid:5) -dimensional initial path and an intersection result 53
0
8.1. A preliminary path γ 53
0
8.2. An initial path γ 54
1ε
8.3. An intersection property 56
Chapter 9. Completion of the proof of Theorem 1.3 61
Chapter 10. Proof of Proposition 8.3 63
10.1. An interaction estimate 63
10.2. Preliminary asymptotic estimates 64
10.3. Proof of Proposition 10.1 69
v
vi CONTENTS
Chapter 11. Proof of Lemma 6.1 77
Chapter 12. Generalization to a saddle point setting 83
12.1. Saddle point setting 83
12.2. Proof of Theorem 12.1 84
Acknowledgments 85
Bibliography 87
Abstract
We study the following singularly perturbed problem
−ε2Δu+V(x)u=f(u) in RN.
Our main result is the existence of a family of solutions with peaks that cluster
near a local maximum of V(x). A local variational and deformation argument in
aninfinitedimensionalspaceisdevelopedtoestablishtheexistenceofsuchafamily
forageneralclassofnonlinearitiesf. Earlierworksinthisdirectioncanbefoundin
[KW, DLY, DY, NY]forf(ξ)=ξp (1<p< N+2 whenN ≥3,1<p<∞when
N−2
N =1,2). These papers use the Lyapunov-Schmidt reduction method, where it is
essential to have information about the null space of the linearization of a solution
of the limit equation −Δu+u = up. Such spectral information is difficult to get
and can only be obtained for very special f(cid:3)s. Our new approach in this memoir
does not require such a detailed knowledge of the spectrum and works for a much
more general class of nonlinearities f.
ReceivedbytheeditorMay18,2012,and,inrevisedform,June27,2012.
ArticleelectronicallypublishedonOctober10,2013.
DOI:http://dx.doi.org/10.1090/memo/1076
2010 MathematicsSubjectClassification. Primary35J60(35B25,35Q55,58E05).
Key words and phrases. Nonlinear Sch¨odinger equations, Singular perturbation, semi-
classicalstandingwaves,localvariationalmethod,interactionestimate,translationflow.
Author affiliations at time of publication: Jaeyoung Byeon, Department of Mathemati-
cal Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea, email:
[email protected]; and Kazunaga Tanaka, Department of Mathematics, School of Science and
Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan, email:
[email protected].
(cid:3)c2013 American Mathematical Society
vii

CHAPTER 1
Introduction and results
In this memoir, we consider the following singular perturbation problem:
⎧
⎨ −ε2Δv+V(x)v =f(v) in RN,
(1.1) v(x)>0 in RN,
⎩
v(x)→0 as |x|→∞
when N ≥3 and ε>0 is small. The equation (1.1) results from studying so-called
semiclassical standing waves, i.e. solutions of the form exp(−iEt/(cid:2))v(x) for the
nonlinear Schro¨dinger equation
∂ψ (cid:2)2
i(cid:2) + Δψ−V(x)ψ+f(ψ)=0, (t,x)∈R×RN.
∂t 2
This evolution equation describes the dynamics of a group of identical particles
interacting with each other in ultra-cold states, in particular, Bose-Einstein con-
densates (BEC) via the Hartree approximation, to an excellent degree of accuracy.
Infact,sincethetheoreticpredictionofBECbyBoseandEinsteininthe1920’sand
an experimental verification of BEC by Cornell and Wieman in 1995, there have
been many efforts to get rigorous models of BEC using the nonlinear Schro¨dinger
equation starting from the many-body Hamiltonian of interacting particles (see
[LS] and [M] for some related materials). It is well known that the equation also
arises in many other fields of physics (see [M] and references therein).
The pioneering work [FW] of Floer and Weinstein is the first research we
know of on (1.1). They studied the existence of one peak solutions. A one peak
solution is a solution that concentrates at a point as ε → 0. In fact, Floer and
Weinstein showed by the Lyapunov-Schmidt reduction method that when N =
1,V ∈C2(RN),V(cid:3)(x )=0,V(cid:3)(cid:3)(x )(cid:7)=0, i.e. x is a nondegenerate local maximum
0 0 0
orminimumpointofV,andf(u)=u3,thereexistsapositivesolutionu ∈H1(R)
ε
of(1.1). Thesesolutionshavethefollowingproperties: foramaximumpointx ∈R
ε
of uε, limε→0xε =x0 and uε(ε·+xε) converges uniformly to Ux0 as ε →0, where
U is the unique positive solution of
x0
U(cid:3)(cid:3)−V(x )U +U3 =0 in R, U(x)=U(−x), x∈R.
0
Subsequently, also using the Lyapunov-Schmidt reduction method, further exten-
sions of this work to higher dimensional cases and more general types of critical
points of V have been obtained by many authors in [ABC,AMN,DKW,KW,
L,O1,O2] (see also references therein). These papers require a linearized non-
degeneracycondition, namely,thatforacriticalpointx ∈RN ofV andapositive
0
solution U ∈H1(RN) of an autonomous limiting problem
ΔU −V(x )U +f(U)=0 in RN,
0
1