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
