Perturbation methods and semilinear elliptic problems on R[superscript n] / Antonio Ambrosetti, Andrea Malchiodi.

Format:

Book

Author/Creator:

Ambrosetti, A. (Antonio)

Contributor:

Malchiodi, Andrea.

Series:

Progress in mathematics (Boston, Mass.) ; v. 240.

Progress in mathematics ; v. 240

Language:

English

Subjects (All):

Differential equations, Elliptic.

Perturbation (Mathematics).

Boundary value problems.

Physical Description:

ix, 183 pages : illustrations ; 24 cm.

Place of Publication:

Basel ; Boston : Birkhäuser Verlag, [2006]

Summary:

This book has been awarded the Ferran Sunyer i Balaguer 2005 prize.

The aim of this monograph is to discuss several elliptic problems on Rn with two main features: they are variational and perturbative in nature, and standard tools of nonlinear analysis based on compactness arguments cannot be used in general. For these problems, a more specific approach that takes advantage of such a perturbative setting seems to be the most appropriate. The first part of the book is devoted to these abstract tools, which provide a unified frame for several applications, often considered different in nature. Such applications are discussed in the second part, and include semilinear elliptic problems on Rn, bifurcation from the essential spectrum, the prescribed scalar curvature problem, nonlinear Schrdinger equations, and singularly perturbed elliptic problems in domains. These topics are presented in a systematic and unified way.

Contents:

1 Examples and Motivations

1.1 Elliptic equations on R[superscript n] 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[subscript epsilon] 29

2.3 Critical points for a class of perturbed functionals, II 29

2.4 A more general case 33

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 R[superscript n] with Subcritical Growth

4.1 The abstract setting 45

4.2 Study of the Ker[I''[subscript 0](z[subscript xi])] 47

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

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

8 Nonlinear Schrodinger 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

9.2 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'[subscript epsilon](z + w) = 0 156

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[subscript epsilon]) 162

10.5.1 The finite-dimensional reduction 163

10.5.2 Proof of Theorem 10.12 166

10.5.3 Further results 171.

Notes:

Includes bibliographical references (pages [173]-180) and index.

ISBN:

3764373210

OCLC:

62325150

Publisher Number:

9783764373214