Schechter Minimax Systems and Critical Point Theory (Springer, 2009)

Martin Schechter

Minimax Systems

and Critical Point Theory

Birkhäuser

Boston • Basel • Berlin

Martin Schechter Mathematics Department University of California Irvine, CA 92697-3875 [e-mail]

ISBN 978-0-8176-4805-3	e-ISBN 978-0-8176-4902-9
DOI 10.1007/ 978-0-8176-4902-9

Library of Congress Control Number: 2009928827

Mathematics Subject Classification (2000): 35J20, 35J65, 47J30, 49J35, 58E05

© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009

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BS D

To my wife, Deborah, our children, our grandchildren (twenty four so far) our great grandchildren (six so far) and our extended family.

May they all enjoy many happy years.

Contents

Preface			xi
1 Critical Points of Functionals			1
	1.1	Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	1
	1.2	Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	2
	1.3	Palais–Smale sequences . . . . . . . . . . . . . . . . . . . . . . . . .	2
	1.4	Cerami sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . .	3
	1.5	Linking sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	3
	1.6	Previous definitions of linking . . . . . . . . . . . . . . . . . . . . .	4
	1.7	Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .	5
2	Minimax Systems		7
	2.1	Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	7
	2.2	Definitions and theorems . . . . . . . . . . . . . . . . . . . . . . . .	8
	2.3	Linking subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	9
	2.4	A variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	11
	2.5	Weaker conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .	12
	2.6	Some consequences . . . . . . . . . . . . . . . . . . . . . . . . . . .	13
	2.7	Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .	15
3 Examples of Minimax Systems			17
	3.1	Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	17
	3.2	A method using homeomorphisms . . . . . . . . . . . . . . . . . . .	17
	3.3	A method using metric spaces . . . . . . . . . . . . . . . . . . . . .	19
	3.4	A method using homotopy-stable families . . . . . . . . . . . . . . .	19
	3.5	Examples of linking sets . . . . . . . . . . . . . . . . . . . . . . . .	21
	3.6	Various geometries . . . . . . . . . . . . . . . . . . . . . . . . . . .	24
	3.7	A sandwich theorem . . . . . . . . . . . . . . . . . . . . . . . . . .	26
	3.8	Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .	29
4	Ordinary Differential Equations		31
	4.1	Extensions of Picard’s theorem . . . . . . . . . . . . . . . . . . . . .	31
	4.2	Estimating solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .	33
	4.3	Extending solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .	34

viii			Contents
	4.4	The proofs . . . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	35
	4.5	An important estimate .	. . . . . . . . . . . . . . . . . . . . . . . . .	36
5	The Method Using Flows			39
	5.1	Introduction . . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	39
	5.2	Theorem 2.4 . . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	39
	5.3	Theorem 2.12 . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	41
	5.4	Theorem 2.14 . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	44
	5.5	Theorem 2.21 . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	45
6	Finding Linking Sets			51
	6.1	Introduction . . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	51
	6.2	The strong case . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	52
	6.3	The remaining proofs .	. . . . . . . . . . . . . . . . . . . . . . . . .	54
	6.4	Notes and remarks . .	. . . . . . . . . . . . . . . . . . . . . . . . .	56
7	Sandwich Pairs			57
	7.1	Introduction . . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	57
	7.2	Criteria . . . . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	58
	7.3	Notes and remarks . .	. . . . . . . . . . . . . . . . . . . . . . . . .	61
8	Semilinear Problems			63
	8.1	Introduction . . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	63
	8.2	Bounded domains . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	63
	8.3	Some useful quantities	. . . . . . . . . . . . . . . . . . . . . . . . .	69
	8.4	Unbounded domains .	. . . . . . . . . . . . . . . . . . . . . . . . .	71
	8.5	Further applications . .	. . . . . . . . . . . . . . . . . . . . . . . . .	75
	8.6	Special cases . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	80
	8.7	The proofs . . . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	81
	8.8	Notes and remarks . .	. . . . . . . . . . . . . . . . . . . . . . . . .	83
9	Superlinear Problems			85
	9.1	Introduction . . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	85
	9.2	The main theorems . .	. . . . . . . . . . . . . . . . . . . . . . . . .	86
	9.3	Preliminaries . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	88
	9.4	Proofs . . . . . . . . .	. . . . . . . . . . . . . . . . . . . . . . . . .	89
	9.5	The parameter problem	. . . . . . . . . . . . . . . . . . . . . . . . .	92
	9.6	The monotonicity trick	. . . . . . . . . . . . . . . . . . . . . . . . .	97
	9.7	Notes and remarks . .	. . . . . . . . . . . . . . . . . . . . . . . . .	104
10	Weak Linking			107

10.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

10.2Another norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

10.3	Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	112
10.4	Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . .	114
10.5	Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .	126

Contents	ix
11 Fuc´ıkˇ Spectrum: Resonance	129

11.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

11.2The curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

11.3	Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	135
11.4	Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .	139
12 Rotationally Invariant Solutions		141

12.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

12.2The spectrum of the linear operator . . . . . . . . . . . . . . . . . . . 142

12.3	The nonlinear case . . . . . . . . . . . . . . . . . . . . . . . . . . .	144
12.4	Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .	147
13 Semilinear Wave Equations		149

13.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

13.2Convexity and lower semi-continuity . . . . . . . . . . . . . . . . . . 149

13.3 Existence of saddle points . . . . . . . . . . . . . . . . . . . . . . . 152

13.4Criteria for convexity . . . . . . . . . . . . . . . . . . . . . . . . . . 155

13.5Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

	13.6	The theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	158
	13.7	The proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	158
	13.8	Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .	161
14	Type (II) Regions		163
	14.1	Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	163
	14.2	The asymptotic equation . . . . . . . . . . . . . . . . . . . . . . . .	166
	14.3	Local estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	168
	14.4	The solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	172
	14.5	Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .	175
15 Weak Sandwich Pairs			177
	15.1	Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	177
	15.2	Weak sandwich pairs . . . . . . . . . . . . . . . . . . . . . . . . . .	178
	15.3	Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	184
	15.4	Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .	190
16	Multiple Solutions		191

16.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

16.2Two examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

16.3	Statement of the theorems . . . . . . . . . . . . . . . . . . . . . . .	192
16.4	Some lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	194
16.5	Local linking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	206
16.6	The proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	208
16.7	Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .	209

x		Contents
17 Second-Order Periodic Systems			213
17.1	Introduction . . . . . . . . . . .	. . . . . . . . . . . . . . . . . . . .	213
17.2	Proofs of the theorems . . . . .	. . . . . . . . . . . . . . . . . . . .	215
17.3	Nonconstant solutions . . . . . .	. . . . . . . . . . . . . . . . . . . .	221
17.4	Notes and remarks . . . . . . .	. . . . . . . . . . . . . . . . . . . .	226
Bibliography			229
Index			241

Preface

Many problems in science involve the solving of differential equations or systems of differential equations. Moreover, many of these equations and systems come from variational considerations involving mappings (called functionals) into the real number system. As a simple example, consider the problem of finding a solution of

(1)	u (x) = f (x, u(x)), x I = [a, b],
under the conditions
(2)	u(a) = u(b) = 0.

Assume that the function f (x, t) is continuous in I × R. The corresponding functional is

			b	)2 + 2F(x, u)] dx,
(3)	G(u) =	a	[(u

where

t

F(x, t) := f (x, s) ds.

0

It is easy to show that u(x) is a solution of the problem (1), (2) if and only if it satisfies

(4)	G (u) = 0.

Thus, in such cases, solutions of the equations or systems are critical points of the corresponding functional. As a result, anyone who is interested in obtaining solutions of the equations or systems is also interested in obtaining critical points of the corresponding functionals. The latter problem is the subject of this book.

The classical way of obtaining critical points was to search for maxima or minima. This is possible if the functional is bounded from above or below. However, when this is not the case, there is no organized way of finding critical points.

Linking theory is an attempt to “level the playing field,” i.e., to find a substitute for semiboundedness. It finds a pair of subsets A, B of the underlying space that allow the functional to have the same advantages as semibounded functionals if the subsets separate the functional. They separate a functional G if

(5)	sup G < inf G.

A B