Ambrosetti / Malchiodi Perturbation Methods and Semilinear Elliptic Problems on R^n

1. Auflage 2006
ISBN: 978-3-7643-7396-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 240, 184 Seiten

Reihe: Progress in Mathematics

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.

Autoren/Hrsg.

Ambrosetti, Antonio

Malchiodi, Andrea

Inhaltsverzeichnis

1;Contents;7
2;Foreword;10
2.1;Notation;11
3;1 Examples and Motivations;12
3.1;1.1 Elliptic equations on Rn;12
3.2;1.2 Bifurcation from the essential spectrum;16
3.3;1.3 Semiclassical standing waves of NLS;17
3.4;1.4 Other problems with concentration;19
3.5;1.5 The abstract setting;21
4;2 Pertubation in Critical Point Theory;24
4.1;2.1 A review on critical point theory;24
4.2;2.2 Critical points for a class of perturbed functionals, I;30
4.3;2.3 Critical points for a class of perturbed functionals, II;40
4.4;2.4 A more general case;44
5;3 Bifurcation from the Essential Spectrum;46
5.1;3.1 A first bifurcation result;46
5.2;3.2 A second bifurcation result;50
5.3;3.3 A problem arising in nonlinear optics;52
6;4 Elliptic Problems on Rn with Subcritical Growth;56
6.1;4.1 The abstract setting;56
6.2;4.2 Study of the;58
6.3;4.3 A .rst existence result;61
6.4;4.4 Another existence result;63
7;5 Elliptic Problems with Critical Exponent;70
7.1;5.1 The unperturbed problem;70
7.2;5.2 On the Yamabe-like equation;73
7.3;5.3 Further existence results;79
8;6 The Yamabe Problem;84
8.1;6.1 Basic notions and facts;84
8.2;6.2 Some geometric preliminaries;87
8.3;6.3 First multiplicity results;91
8.4;6.4 Existence of infinitely-many solutions;99
8.5;6.5 Appendix;103
9;7 Other Problems in Conformal Geometry;112
9.1;7.1 Prescribing the scalar curvature of the sphere;112
9.2;7.2 Problems with symmetry;116
9.3;7.3 Prescribing Scalar and Mean Curvature on manifolds with boundary;120
10;8 Nonlinear Schrödinger Equations;126
10.1;8.1 Necessary conditions for existence of spikes;126
10.2;8.2 Spikes at non-degenerate critical points of V;128
10.3;8.3 The general case: Preliminaries;132
10.4;8.4 A modified abstract approach;134
10.5;8.5 Study of the reduced functional;142
11;9 Singularly Perturbed Neumann Problems;146
11.1;9.1 Preliminaries;146
11.2;9.2 Construction of approximate solutions;149
11.3;9.3 The abstract setting;154
11.4;9.4 Proof of Theorem 9.1;157
12;10 Concentration at Spheres for Radial Problems;162
12.1;10.1 Concentration at spheres for radial NLS;162
12.2;10.2 The finite-dimensional reduction;164
12.3;10.3 Proof of Theorem 10.1;170
12.4;10.4 Other results;171
12.5;10.5 Concentration at spheres for Ne;173
13;Bibliography;184
14;Index;192

Leseproben

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.