Perturbation Methods and Semilinear Elliptic Problems on R^n

ISBN-13: 9783764373214 / Angielski / Twarda / 2005 / 184 str.

Antonio Ambrosetti; Andrea Malchiodi

Opis

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

Spis treści

Examples and Motivations.- Pertubation in Critical Point Theory.- Bifurcation from the Essential Spectrum.- Elliptic Problems on ?n with Subcritical Growth.- Elliptic Problems with Critical Exponent.- The Yamabe Problem.- Other Problems in Conformal Geometry.- Nonlinear Schrödinger Equations.- Singularly Perturbed Neumann Problems.- Concentration at Spheres for Radial Problems.

Blurb

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 Schrödinger equations, and singularly perturbed
elliptic problems in domains. These topics are presented in a systematic and
unified way.