Perturbation Methods and Semilinear Elliptic Problems on R^n

187 Pages·2006·1.074 MB·English

by  Antonio Ambrosetti, Andrea Malchiodi

Progress in Mathematics
Volume 240
Series Editors
H. Bass
J. Oesterlé
A. Weinstein
Antonio Ambrosetti
Andrea Malchiodi
Perturbation Methods
and Semilinear Elliptic
Problems on R n

Birkhäuser Verlag
Basel (cid:1) Boston (cid:1) Berlin
Authors:
Antonio Ambrosetti
Andrea Malchiodi
S.I.S.S.A.
Via Beirut 2-4
34014 Trieste
Italy
e-mail: [email protected]
e-mail: [email protected]
2000 Mathematics Subject Classifi cation 34A47, 35A20, 35J60, 35Q55, 53A30
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografi e;
detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 3-7643-7321-0 Birkhäuser Verlag, Basel – Boston – Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifi cally the rights of translation, reprinting, re-use of
illustrations, broadcasting, reproduction on microfi lms or in other ways, and storage in
data banks. For any kind of use whatsoever, permission from the copyright owner must
be obtained.
© 2006 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland
Part of Springer Science+Business Media
Printed on acid-free paper produced of chlorine-free pulp. TCF ∞
Printed in Germany
ISBN-10: 3-7643-7321-0 e-ISBN: 3-7643-7396-2
ISBN-13: 978-3-7643-7321-4
9 8 7 6 5 4 3 2 1 www.birkhauser.ch
Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1 Examples and Motivations
1.1 Elliptic equations on Rn . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The subcritical case . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 The critical case: the Scalar Curvature Problem. . . . . . . 3
1.2 Bifurcation from the essential spectrum . . . . . . . . . . . . . . . 5
1.3 Semiclassical standing waves of NLS . . . . . . . . . . . . . . . . . 6
1.4 Other problems with concentration . . . . . . . . . . . . . . . . . . 8
1.4.1 Neumann singularly perturbed problems . . . . . . . . . . . 8
1.4.2 Concentration on spheres for radial problems . . . . . . . . 9
1.5 The abstract setting . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Pertubation in Critical Point Theory
2.1 A review on critical point theory . . . . . . . . . . . . . . . . . . . 13
2.2 Critical points for a class of perturbed functionals, I . . . . . . . . 19
2.2.1 A finite-dimensional reduction:
the Lyapunov-Schmidt method revisited . . . . . . . . . . . 20
2.2.2 Existence of critical points. . . . . . . . . . . . . . . . . . . 22
2.2.3 Other existence results . . . . . . . . . . . . . . . . . . . . . 24
2.2.4 A degenerate case . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.5 A further existence result . . . . . . . . . . . . . . . . . . . 27
2.2.6 Morse index of the critical points of I . . . . . . . . . . . . 29
ε
2.3 Critical points for a class of perturbed functionals, II . . . . . . . . 29
2.4 A more general case . . . . . . . . . . . . . . . . . . . . . . . . . . 33
viii Contents
3 Bifurcation from the Essential Spectrum
3.1 A first bifurcation result . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 The unperturbed problem . . . . . . . . . . . . . . . . . . . 36
3.1.2 Study of G . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 A second bifurcation result . . . . . . . . . . . . . . . . . . . . . . 39
3.3 A problem arising in nonlinear optics . . . . . . . . . . . . . . . . . 41
4 Elliptic Problems on Rn with Subcritical Growth
4.1 The abstract setting . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Study of the Ker[I(cid:1)(cid:1)(z )] . . . . . . . . . . . . . . . . . . . . . . . . 47
0 ξ
4.3 A first existence result . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Another existence result . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Elliptic Problems with Critical Exponent
5.1 The unperturbed problem . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 On the Yamabe-like equation . . . . . . . . . . . . . . . . . . . . . 62
5.2.1 Some auxiliary lemmas . . . . . . . . . . . . . . . . . . . . 63
5.2.2 Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . 66
5.2.3 The radial case . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3 Further existence results . . . . . . . . . . . . . . . . . . . . . . . . 68
6 The Yamabe Problem
6.1 Basic notions and facts . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.1.1 The Yamabe problem . . . . . . . . . . . . . . . . . . . . . 74
6.2 Some geometric preliminaries . . . . . . . . . . . . . . . . . . . . . 76
6.3 First multiplicity results . . . . . . . . . . . . . . . . . . . . . . . . 80
6.3.1 Expansions of the functionals . . . . . . . . . . . . . . . . . 80
6.3.2 The finite-dimensional functional . . . . . . . . . . . . . . . 82
6.3.3 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . 86
6.4 Existence of infinitely-many solutions. . . . . . . . . . . . . . . . . 88
6.4.1 Proof of Theorem 6.3 completed . . . . . . . . . . . . . . . 90
6.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 Other Problems in Conformal Geometry
7.1 Prescribing the scalar curvature of the sphere . . . . . . . . . . . . 101
7.2 Problems with symmetry . . . . . . . . . . . . . . . . . . . . . . . 105
7.2.1 The perturbative case . . . . . . . . . . . . . . . . . . . . . 105
7.3 Prescribing Scalar and Mean Curvature
on manifolds with boundary . . . . . . . . . . . . . . . . . . . . . . 109
7.3.1 The Yamabe-like problem . . . . . . . . . . . . . . . . . . . 109
7.3.2 The Scalar Curvature Problem with
boundary conditions . . . . . . . . . . . . . . . . . . . . . . 111
Contents ix
8 Nonlinear Schro¨dinger Equations
8.1 Necessary conditions for existence of spikes . . . . . . . . . . . . . 115
8.2 Spikes at non-degenerate critical points of V . . . . . . . . . . . . . 117
8.3 The general case: Preliminaries . . . . . . . . . . . . . . . . . . . . 121
8.4 A modified abstract approach . . . . . . . . . . . . . . . . . . . . . 123
8.5 Study of the reduced functional . . . . . . . . . . . . . . . . . . . . 131
9 Singularly Perturbed Neumann Problems
9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.2 Construction of approximate solutions . . . . . . . . . . . . . . . . 138
9.3 The abstract setting . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.4 Proof of Theorem 9.1. . . . . . . . . . . . . . . . . . . . . . . . . . 146
10 Concentration at Spheres for Radial Problems
10.1 Concentration at spheres for radial NLS . . . . . . . . . . . . . . . 151
10.2 The finite-dimensional reduction . . . . . . . . . . . . . . . . . . . 153
10.2.1 Some preliminary estimates . . . . . . . . . . . . . . . . . . 154
10.2.2 Solving PI(cid:1)(z+w)=0 . . . . . . . . . . . . . . . . . . . . 156
ε
10.3 Proof of Theorem 10.1 . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.3.1 Proof of Theorem 10.1 completed . . . . . . . . . . . . . . . 160
10.4 Other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
10.5 Concentration at spheres for (N ) . . . . . . . . . . . . . . . . . . . 162
ε
10.5.1 The finite-dimensional reduction . . . . . . . . . . . . . . . 163
10.5.2 Proof of Theorem 10.12 . . . . . . . . . . . . . . . . . . . . 166
10.5.3 Further results . . . . . . . . . . . . . . . . . . . . . . . . . 171
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Foreword
Several important problems arising in Physics, Differential Geometry and other
topics lead to consider semilinear variational elliptic equations on Rn and a great
deal of work has been devoted to their study. From the mathematical point of
view, the main interest relies on the fact that the tools of Nonlinear Functional
Analysis, based on compactness arguments,in general cannot be used, at least in
a straightforwardway, and some new techniques have to be developed.
On the other hand, there are several elliptic problems on Rn which are per-
turbative in nature. In some cases there is a natural perturbation parameter, like
inthebifurcationfromtheessentialspectrumorinsingularlyperturbedequations
or in the study of semiclassical standing waves for NLS. In some other circum-
stances, one studies perturbations either because this is the first step to obtain
global results or else because it often provides a correct perspective for further
global studies.
Fortheseperturbationproblemsaspecificapproach,thattakesadvantageof
such a perturbative setting, seems the most appropriate.These abstract tools are
provided by perturbation methods in critical point theory. Actually, it turns out
that such a frameworkcan be used to handle a large variety of equations,usually
considered different in nature.
Theaimofthismonographistodiscusstheseabstractmethodstogetherwith
their applications to several perturbation problems, whose common feature is to
involvesemilinear Elliptic PartialDifferential Equations on Rn with a variational
structure.
The resultspresentedhereare basedonpapersofthe Authors carriedout in
the last years.Many of them are works in collaborationwith other people like D.
Arcoya,M. Badiale,M. Berti,S. Cingolani,V. Coti Zelati,J.L. Gamez, J. Garcia
Azorero, V. Felli, Y.Y. Li, W.M. Ni, I. Peral, S. Secchi. We would like to express
our warm gratitude to all of them.
Notation
• Rn is the Euclidean n-dimensional space with points x=(x ,...,x ).
1 n
• (cid:1)x,y(cid:2) denote the Euclidean scalar product of x,y ∈ Rn; we also set |x|2 =
(cid:1)x,x(cid:2).
• B (y) is the ball {x∈Rn :|x−y|<r}. We will write B to shorten B (0).
r r r
• Sn denotes the unit n-dimensional sphere: Sn ={x∈Rn+1 :|x|=1}.
• If Ω is an open subset of Rn and u : Ω (cid:4)→ R is smooth, we denote by D u,
i
D2u the partial derivatives of u with respect to x , x x , etc.; we will also
ij i i j
use the notation ∂ or∂ insteadofD ,and ∂2 or∂2 insteadofD2.
∂xi xi i ∂xi∂xj xixj ij
• ∇u denotes the gradient of real-valued function u: ∇u = (D u,...,D u);
1 n
sometime, for a real-valued function K, the notation K(cid:1) will also be used
instead of ∇K.
• ∇u·∇v will be also used to denote (cid:1)∇u,∇v(cid:2).
(cid:1)
• ∆ denotes the Laplacian: ∆u= n ∂2 .
1 ∂x2
• Ifu,v ∈H,a(real)Hilbertspace,thescialarproductwillbedenotedby(u|v)
and the norm (cid:7)u(cid:7)2 =(u|u).
• Id denotes the identity map in Rn or H.
• Lp(Rn), Lp (Rn), Lp(Ω), etc. denote the usual Lebesgue spaces.
loc
• Wm,p(Rn),Wm.p(Ω),etc.denotetheusualSobolevspaces.IfM isasmooth
manifold, Hm(M) denotes the Sobolev space Hm,2(M).
• 2∗ stands for 2n if n≥3, and 2∗ =+∞ if n=1,2.
n−2
• D1,2(Rn), n≥3, denotes the space {u∈L2∗(Rn):∇u∈L2(Rn)}.
• IfX,Y areBanachspaces,L(X,Y)denotesthespaceofboundedlinearmaps
from X to Y.
• If f ∈ Ck(X,Y), k ≥ 1, df(u), d2f(u), denote the Fr´echet derivatives of f
at u∈X. They are, respectively, a linear bounded map from X to Y, and a
bilinear continuous map fro X×X to Y.
• If I ∈ Ck(H,R), k ≥ 1, is a functional, I(cid:1)(u) denotes the gradient of I
at u ∈ H, defined by means of the Riesz representation Theorem setting
(I(cid:1)(u)|v) = dI(u)[v], ∀v ∈ H. Similarly, I(cid:1)(cid:1)(u) is the linear operator defined
by setting (I(cid:1)(cid:1)(u)v|w)=d2I(u)[v,w], ∀v,w ∈H
• If I ∈C1(H,R), Cr[I] denotes the set of critical points of I.
• u=o(εk) means thatuε −k tends to zero as ε→0.
• u=O(εk) means that|uε −k|≤c as ε→0.
• o (1) denotes a function depending on ε that tends to 0 as ε→0. Similarly,
ε
o (1) denotes a function depending on R that tends to 0 as R→+∞.
R
• The notation ∼ denotes quantities which, in the limit are of the same order.