Perturbation Methods and Semilinear Elliptic Problems on Rⁿ

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 Iε . . . . . . . . . . . . 29

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

2.4 Amore 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 problemarising 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 0 (zξ)] . . . . . . . . . . . . . . . . . . . . . . . . 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 Theorem5.3 . . . . . . . . . . . . . . . . . . . . . 66

5.2.3 The radial case . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Further existence results . . . . . . . . . . . . . . . . . . . . . . . . 68

6 TheYamabeProblem

6.1 Basic notions and facts . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.1.1 The Yamabe problem . . . . . . . . . . . . . . . . . . . . . 74

6.2 Some geometric preliminaries . . . . . . . . . . . . . . . . . . . . . 76

6.3 Firstmultiplicity results . . . . . . . . . . . . . . . . . . . . . . . . 80

6.3.1 Expansions of the functionals . . . . . . . . . . . . . . . . . 80

6.3.2 The finite-dimensional functional . . . . . . . . . . . . . . . 82

6.3.3 Proof of Theorem6.2 . . . . . . . . . . . . . . . . . . . . . 86

6.4 Existence of infinitely-many solutions . . . . . . . . . . . . . . . . . 88

6.4.1 Proof of Theorem6.3 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 Schr¨odinger 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 Amodified 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 Theorem9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 PIε(z + w)=0 . . . . . . . . . . . . . . . . . . . . 156

10.3 Proof of Theorem10.1 . . . . . . . . . . . . . . . . . . . . . . . . . 159

10.3.1 Proof of Theorem 10.1 completed . . . . . . . . . . . . . . . 160

10.4 Other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

10.5 Concentration at spheres for (Nε) . . . . . . . . . . . . . . . . . . . 162

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