An introduction to nonlinear functional analysis and elliptic problems
This book provides some basic abstract tools used in modern nonlinear analysis in strong relationship with their applications to semilinear elliptic boundary value problems. The content of the book is divided into two parts, which contain 13 chapters. In the first part, key results are discussed such as the Banach contraction principle, a fixed point theorem for increasing operators, local and global inversion theory, Leray-Schauder degree, critical point theory, and bifurcation theory. The second part of this volume shows how these abstract results apply to Dirichlet elliptic boundary value problems. A more detailed description is provided in what follows. The first chapter deals with general results, such as Fréchet differentiability, Nemytskij operators, weak and strong solutions of the linear Laplace equation, linear compact operators and their eigenvalues, and Sobolev spaces. Chapter 2 is concerned with the Banach contraction principle and with a fixed point theorem for increasing operators. In Chapter 3, some basic results are discussed: the local inversion theorem, the Hadamard-Caccioppoli global inversion theorem and the case in which the map to be inverted has fold singularities. Chapter 4 deals with the Leray-Schauder topological degree. Variational methods are discussed in the next chapter of this volume. The authors are mainly concerned with minima, the mountain pass theorem and the linking theorem. Chapter 6 is concerned with local and global bifurcation theory. The abstract results collected in the first part of the book are applied in the second part to prove existence and multiplicity results for semilinear elliptic Dirichlet boundary value problems on bounded domains in \({\mathbb R}^N\). The authors point out that the choice of the appropriate abstract tool depends on the behavior of the nonlinearity as well as on the kind of results one expects. In Chapter 7, it is argued how a semilinear elliptic boundary value problem can be transformed into an operator equation in an appropriate Banach or Hilbert function space. In Chapter 8, the case is considered in which the nonlinear term is sublinear at infinity and one can prove \textit{a priori} estimates for possible solutions. Chapter 9 deals with asymptotically linear problems, for which one can also use several different approaches such as global bifurcation or variational methods. In the next chapter, problems with asymmetric nonlinearities are considered, when the behavior at \(+\infty\) and \(-\infty\) is different. Nonlinearities that are superlinear at infinity are considered in Chapter 11 by means of the mountain pass or linking theorems. The last two chapters of the book are concerned with slightly more advanced topics of current research. In Chapter 12, a class of quasilinear elliptic problems is discussed using critical point theory. Here, the corresponding Euler functional is not of class \(C^1\), and hence a new form of the mountain pass theorem has to be proved. Chapter 13 deals with nonlinear Schrödinger equations on the whole Euclidean space. In this setting, the authors establish the existence of ground and bound states as well as semi-classical states. This monograph is suitable for graduate students and researchers who have a good background in nonlinear analysis and have mastered such key techniques as variational methods, critical point theory and weak convergence. This reviewer believes that the volume under review should certainly be in the library of every university where research in mathematics is conducted.
scientific article; zbMATH DE number 5915627