Perturbation Methods and Semilinear Elliptic Problems on R^n (eBook)

Antonio Ambrosetti, Andrea Malchiodi (Autoren)

eBook Download: PDF

2006
XII, 184 Seiten
Springer Basel (Verlag)
978-3-7643-7396-2 (ISBN)

Infos

Several important problems arising in Physics, Di?erential Geometry and other n topics lead to consider semilinear variational elliptic equations on R 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 straightforward way, and some new techniques have to be developed. n On the other hand, there are several elliptic problems on R which are p- turbative in nature. In some cases there is a natural perturbation parameter, like inthe bifurcationfromthe essentialspectrum orinsingularlyperturbed equations or in the study of semiclassical standing waves for NLS. In some other circ- stances, one studies perturbations either because this is the ?rst step to obtain global results or else because it often provides a correct perspective for further global studies. For these perturbation problems a speci?c approach,that takes advantage of 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 framework can be used to handle a large variety of equations, usually considered di?erent in nature. Theaimofthismonographistodiscusstheseabstractmethodstogetherwith their applications to several perturbation problems, whose common feature is to n involve semilinear Elliptic Partial Di?erential Equations on R with a variational structure.

Inhalt

Contents 7
Foreword 10
Notation 11
1 Examples and Motivations 12
1.1 Elliptic equations on Rn 12
1.2 Bifurcation from the essential spectrum 16
1.3 Semiclassical standing waves of NLS 17
1.4 Other problems with concentration 19
1.5 The abstract setting 21
2 Pertubation in Critical Point Theory 24
2.1 A review on critical point theory 24
2.2 Critical points for a class of perturbed functionals, I 30
2.3 Critical points for a class of perturbed functionals, II 40
2.4 A more general case 44
3 Bifurcation from the Essential Spectrum 46
3.1 A first bifurcation result 46
3.2 A second bifurcation result 50
3.3 A problem arising in nonlinear optics 52
4 Elliptic Problems on Rn with Subcritical Growth 56
4.1 The abstract setting 56
4.2 Study of the 58
4.3 A .rst existence result 61
4.4 Another existence result 63
5 Elliptic Problems with Critical Exponent 70
5.1 The unperturbed problem 70
5.2 On the Yamabe-like equation 73
5.3 Further existence results 79
6 The Yamabe Problem 84
6.1 Basic notions and facts 84
6.2 Some geometric preliminaries 87
6.3 First multiplicity results 91
6.4 Existence of infinitely-many solutions 99
6.5 Appendix 103
7 Other Problems in Conformal Geometry 112
7.1 Prescribing the scalar curvature of the sphere 112
7.2 Problems with symmetry 116
7.3 Prescribing Scalar and Mean Curvature on manifolds with boundary 120
8 Nonlinear Schrödinger Equations 126
8.1 Necessary conditions for existence of spikes 126
8.2 Spikes at non-degenerate critical points of V 128
8.3 The general case: Preliminaries 132
8.4 A modified abstract approach 134
8.5 Study of the reduced functional 142
9 Singularly Perturbed Neumann Problems 146
9.1 Preliminaries 146
9.2 Construction of approximate solutions 149
9.3 The abstract setting 154
9.4 Proof of Theorem 9.1 157
10 Concentration at Spheres for Radial Problems 162
10.1 Concentration at spheres for radial NLS 162
10.2 The finite-dimensional reduction 164
10.3 Proof of Theorem 10.1 170
10.4 Other results 171
10.5 Concentration at spheres for Ne 173
Bibliography 184
Index 192

Textauszug

Foreword (P. 11)

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 straightforward way, and some new techniques have to be developed.

On the other hand, there are several elliptic problems on Rn which are perturbative in nature. In some cases there is a natural perturbation parameter, like in the bifurcation from the essential spectrum or in singularly perturbed equations or in the study of semiclassical standing waves for NLS. In some other circumstances, 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.

For these perturbation problems a specific approach, that takes advantage of 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 framework can be used to handle a large variety of equations, usually considered different in nature.

The aim of this monograph is to discuss these abstract methods together with their applications to several perturbation problems, whose common feature is to involve semilinear Elliptic Partial Differential Equations on Rn with a variational structure.

The results presented here are based on papers of the Authors carried out in the last years. Many of them are works in collaboration with 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.

Produkt-Details

• Erscheint lt. Verlag: 21.3.2006
• Reihe/Serie: Progress in Mathematics
• Reihe/Serie: Progress in Mathematics
• Zusatzinfo: XII, 184 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Compactness • differential equation • Equation • Geometry • partial differential equation • Partial differential equations • Perturbation • Semilinear elliptic problems
• ISBN-10: 3-7643-7396-2 / 3764373962
• ISBN-13: 978-3-7643-7396-2 / 9783764373962