Table Of ContentVariational Methods
for Strongly Indefinite
Problems
INTERDISCIPLINARY MATHEMATICAL SCIENCES
Series Editor: Jinqiao Duan (Illinois Inst. of Tech., USA)
Editorial Board: Ludwig Arnold, Roberto Camassa, Peter Constantin,
Charles Doering, Paul Fischer, Andrei V. Fursikov, Fred R. McMorris,
Daniel Schertzer, Bjorn Schmalfuss, Xiangdong Ye, and
Jerzy Zabczyk
Published
Vol. 1: Global Attractors of Nonautonomous Dissipative Dynamical Systems
David N. Cheban
Vol. 2: Stochastic Differential Equations: Theory and Applications
A Volume in Honor of Professor Boris L. Rozovskii
eds. Peter H. Baxendale & Sergey V. Lototsky
Vol. 3: Amplitude Equations for Stochastic Partial Differential Equations
Dirk Blömker
Vol. 4: Mathematical Theory of Adaptive Control
Vladimir G. Sragovich
Vol. 5: The Hilbert–Huang Transform and Its Applications
Norden E. Huang & Samuel S. P. Shen
Vol. 6: Meshfree Approximation Methods with MATLAB
Gregory E. Fasshauer
Vol. 7: Variational Methods for Strongly Indefinite Problems
Yanheng Ding
ZhangJi - Variational Methods.pmd 2 6/26/2007, 9:29 AM
Interdisciplinary Mathematical Sciences – Vol. 7
Variational Methods
for Strongly Indefinite
Problems
Yanheng Ding
Chinese Academy of Sciences, China
World Scientific
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A catalogue record for this book is available from the British Library.
VARIATIONAL METHODS FOR STRONGLY INDEFINITE PROBLEMS
Interdisciplinary Mathematical Sciences — Vol. 7
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-270-962-2
ISBN-10 981-270-962-2
Printed in Singapore.
ZhangJi - Variational Methods.pmd 1 6/26/2007, 9:29 AM
June21,2007 11:27 WorldScienti(cid:12)cBook-9.75inx6.5in VariationalMethods
Preface
This monograph consists of a series of lectures given partly at the Morningside
Center of Mathematics of Chinese Academy of Sciences and the Department of
MathematicsofRutgersUniversity,andentirelyattheDepartmentofMathematics
of the University of Franche-Comt(cid:19)ein a course of nonlinear analysis in March and
April of 2006. The material was mainly taken from some joint work with Thomas
BartschdonewhiletheauthorasanAlexandervonHumboldtfellowvisitedGiessen
University. It presents some results concerning methods in critical point theory
orientedtowardsdi(cid:11)erentialequationswhicharevariationalinnaturewithstrongly
inde(cid:12)nite Lagrangian functionals. The author thanks greatly T. Bartsch for his
kindnesses to him. He would like also to thank H. Br(cid:19)ezis for his encouragements
and F. H. Lin, Y. Y. Li for the discussions on mathematics of common interest.
He also thanks L. Jeanjean for his invitation to come to Besancon and for his
suggestions on the content. Finally he thanks the University of Franche-Comt(cid:19)efor
its optional support.
Yanheng Ding
v
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June21,2007 11:27 WorldScienti(cid:12)cBook-9.75inx6.5in VariationalMethods
Contents
Preface v
1. Introduction 1
2. Lipschitz partitions of unity 5
3. Deformations on locally convex topological vector spaces 15
4. Critical point theorems 25
5. Homoclinics in Hamiltonian systems 35
5.1 Existence and multiplicity results for periodic Hamiltonians . . . . 35
5.2 Spectrum of the Hamiltonian operator . . . . . . . . . . . . . . . . 39
5.3 Variational setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.4 Linking structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.5 The (C) sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.6 Proofs of the main results . . . . . . . . . . . . . . . . . . . . . . . 53
5.7 Non periodic Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 54
5.7.1 Variational setting . . . . . . . . . . . . . . . . . . . . . . . 56
5.7.2 Linking structure . . . . . . . . . . . . . . . . . . . . . . . 60
5.7.3 The (C)-condition . . . . . . . . . . . . . . . . . . . . . . . 62
5.7.4 Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . 65
6. Standing waves of nonlinear Schro(cid:127)dinger equations 67
6.1 Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 The linking structure . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.4 The (C) sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.5 Proofs of the existence and multiplicity . . . . . . . . . . . . . . . 81
6.6 Semiclassical states of a system of Scho(cid:127)dinger equations . . . . . . 82
vii
June21,2007 11:27 WorldScienti(cid:12)cBook-9.75inx6.5in VariationalMethods
viii Variational Methods for Strongly Inde(cid:12)nite Problems
6.6.1 An equivalent variational problem . . . . . . . . . . . . . . 84
6.6.2 Proofs of Theorem 6.5. . . . . . . . . . . . . . . . . . . . . 88
6.6.3 Proof of Theorem 6.6 . . . . . . . . . . . . . . . . . . . . . 93
7. Solutions of nonlinear Dirac equations 97
7.1 Relative studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Existence results for scalar potentials . . . . . . . . . . . . . . . . . 100
7.3 Variational setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.4 The asymptotically quadratic case . . . . . . . . . . . . . . . . . . 106
7.5 Super-quadratic case . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.6 More general external (cid:12)elds . . . . . . . . . . . . . . . . . . . . . . 121
7.6.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.6.2 Variational arguments . . . . . . . . . . . . . . . . . . . . . 123
7.6.3 Proof of Theorem 7.8 . . . . . . . . . . . . . . . . . . . . . 131
7.6.4 Proofs of Theorems 7.6 and 7.7 . . . . . . . . . . . . . . . 131
7.7 Semiclassical solutions . . . . . . . . . . . . . . . . . . . . . . . . . 133
8. Solutions of a system of di(cid:11)usion equations 139
8.1 Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.3 Linear preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.4 Functional setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.5 Solutions to (FS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.6 Some extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.6.1 0 is a boundary point of (cid:27)(S) . . . . . . . . . . . . . . . . 154
8.6.2 More general symmetries . . . . . . . . . . . . . . . . . . . 155
8.6.3 More general nonlinearities . . . . . . . . . . . . . . . . . . 155
8.6.4 More general systems . . . . . . . . . . . . . . . . . . . . . 156
Bibliography 161
Index 167
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Chapter 1
Introduction
The classical Calculus of Variations deals with (cid:12)nding minima of functionals
(cid:8) : X R that are bounded below. The basic idea of the direct method is to
!
consider a minimizing sequence (cid:8)(u ) inf(cid:8), to (cid:12)nd a convergent subsequence
n
!
u u, and to show that (cid:8)(u) = inf(cid:8). In order to make this work the space
nk !
X should have a topology which is rather weak for the existence of a convergent
subsequence, and rather strong so that (cid:8) is lower semicontinuous. In many ap-
plications the functional is not bounded below and instead of a minimizer one is
interested in critical points. This is the concern of the Calculus of Variations in
the Large or Critical Point Theory, which has undergone an enormous develop-
ment in the last century due to the work of mathematicians like Morse, Lusternik,
Schnirelman, Palais, Smale, Rabinowitz, Ambrosetti, Lions, Struwe, Witten, Floer
andmanyothers,withapplicationstoproblemsfromanalysis,geometryandmath-
ematical physics. Here one usually requires X to be a Banach manifold and (cid:8) to
be di(cid:11)erentiable. An essential ingredient is the construction of a (cid:13)ow ’ on X so
that (cid:8)(’(t;u)) is decreasing in t. This (cid:13)ow is used in the spirit of Morse theory,
to construct deformations of sublevel sets (cid:8)c = u X : (cid:8)(u) c , and to (cid:12)nd
f 2 (cid:20) g
Palais-Smalesequences(u ) ,thatis: (cid:8)(u )isboundedand(cid:8)(u ) 0,replacing
n n n 0 n
!
the minimizing sequences. Typical results are the mountain pass theorem of Am-
brosettiandRabinowitzorvariouslinkingtheorems. Theproofsuseinanessential
way topological concepts based on the Brouwer or Leray-Schauder degree. The
theory has also been extended to deal with (semi-)continuous functions on metric
spaces,forcedbyproblemsfromnonlinearelasticity(see [DegiovanniandSchuricht
(1998)]). Another generalization concerns variational methods for functionals on
closed convex subsets of Banach spaces developed by Struwe [Struwe (1989)] for
Plateau’s problem. Such functionals appear also in variational inequalities.
Motivatedbyseveralapplications,forinstanceto(cid:12)nite-andin(cid:12)nite-dimensional
Hamiltonian systems, nonlinear Schro(cid:127)dinger equations and nonlinear Dirac equa-
tions, we were led to consider C1-functionals (cid:8) : E = E E+ R de(cid:12)ned on
(cid:0)
(cid:8) !
the product E =E E+ of Banach spaces E with dimE = but where one
(cid:0) (cid:6) (cid:6)
(cid:8) 1
needs to work with the weak topology on E in order to gain compactness. The
(cid:0)
1
