Preface

This book is an introduction to the study of a wide class of partial differential equations, namely semilinear elliptic problems, through the application of variational methods and critical point theory. Semilinear elliptic equations arise in a variety of contexts in geometry, physics, mechanics, engineering and, more recently, in life sciences. None of these fields can be investigated without taking into account nonlinear phenomena, and variational methods, as a branch or an evolution of the Calculus of Variations, are almost entirely concerned with nonlinearity. The theory has attained a spectacular success in the last thirty-or-so years, reaching a high level of complexity and refinement, and its applications are scattered throughout thousands of research papers. Furthermore, some of the simplest methods within the variational approach are nowadays classical tools in the field of nonlinear differential equations as fixed point and monotonicity methods are. They have become, in other words, part of the toolbox that any researcher in the field is supposed to have at hand. This poses a stimulating problem: how to teach variational methods and semilinear elliptic equations to upcoming generations of students. In our teaching experience, we have noticed a recurrent phenomenon. Up to a certain grade, students possess a rather general knowledge of basic principles, for instance in Functional Analysis. At the next step however they are often confronted with problems taken from real research. This creates a gap that students generally fill in by themselves, following a path of hard work that requires a great deal of commitment and selfdiscipline. Both of us have been teaching courses on the topics collected in this book for a number of years in various Italian universities, and the didactic demand that we encountered was almost always located in the gap described above. We found that the many excellent existing books on variational methods and critical point theory (e.g. [2, 18, 26, 35, 43, 45, 48]), all of them deservedly well-known internationally, are sometimes problematic for classroom use, precisely because they are too complete, or too long, or too advanced. Hence, the purpose of this book, derived directly from classroom experience, is that of providing a support to the students and the teacher engaged in a first course in semilinear elliptic equations. We have tried to write a textbook that matches what