Spectral theory and nonlinear functional analysis (Q2715719)

Bifurcation theory is one of the most important (and oldest) fields of nonlinear analysis. Roughly speaking, in this theory one is interested in the topological structure of the solution set of the equation NEWLINE\[NEWLINEL(\lambda)u+ N(\lambda,u)= 0,\tag{\(*\)}NEWLINE\]NEWLINE where \(L(\lambda)\) is a family of linear operators depending on a parameter \(\lambda\), and \(N(\lambda,\cdot)\) is some nonlinear perturbation depending on the same parameter. In the most classical setting, this dependence is simply \(L(\lambda)=\lambda I- L\), with \(L\) being a fixed bounded linear operator, providing in this way a connection to linear spectral theory. In the nonlinear case the first fundamental bifurcation theorems have been obtained by Krasnosel'skij in the local and by Rabinowitz in the global setting.NEWLINENEWLINENEWLINEIn this book the author studies the equation \((*)\) from several points of view, with a special emphasis on multiplicity results for (nonlinear) eigenvalues, in particular, for Fredholm operators of index zero. The book consists of seven chapters. The first chapter is introductory, while the second chapter is concerned with bifurcation from simple eigenvalues. (Surprisingly, the author ignores the important contributions by T. Küppers and C. A. Stuart in this chapter.) Global bifurcation theorems (e.g., by Ize and Rabinowitz) are considered in the fourth chapter, while the following two chapters are devoted to a detailed discussion of several notions of multiplicity. Finally, in the seventh chapter the author studies some applications to semilinear elliptic equations and elliptic systems.NEWLINENEWLINENEWLINEThis monograph is certainly a valuable contribution to the literature on topological methods of nonlinear analysis. It will be useful for specialists in nonlinear functional analysis and nonlinear partial differential equations. However, the reviewer finds the title extremely misleading: this is definitely not a textbook on nonlinear analysis, nor on spectral theory, nor on the intersection of these two fields, but simply a fairly special monograph on bifurcation theory for abstract nonlinear operators.