The main purpose of nonlinear functional analysis is to develop abstract topological and variational methods to study nonlinear phenomena arising in applications. Although this is a rather recent field, initiated about one hundred years ago, remarkable advances have been made and there are now many results that are well established. The fundamental tools of the Leray–Schauder topological degree, local and global bifurcation and critical point theory, can be considered topics that any graduate student in mathematics and physics should know.

This book discusses a selection of the most basic results dealing with the aforementioned topics. The material is presented as simply as possible, in order to highlight the main ideas. In many cases we prefer to state results under slightly stronger assumptions, when this makes the exposition much more clear and avoids some unnecessary technicalities.

The abstract tools are discussed taking into account their applications to semilinear elliptic problems. In some sense, elliptic equations become like a guiding thread, along which the reader will recognize how one method is more suitable than another one, according to the specific feature of the nonlinearity. This is the reason why we discuss both topological methods and variational tools.

After a first chapter containing preliminary material, the book is divided into four parts. The first part is devoted to topological methods and bifurcation theory. Chapter 2 deals with the Lyapunov–Schmidt reduction method and the bifurcation from a simple eigenvalue and connects with the previous book A Primer of Nonlinear Analysis [20], of which the present book is a follow up. Chapter 3 deals with the topological degree.