Antonio Ambrosetti Livres

Focusing on nonlinear differential equations, this graduate text explores advanced methods of nonlinear analysis through topological and variational concepts. It introduces essential techniques such as bifurcation theory and critical point theory, while also addressing elliptic partial differential equations. The book includes well-illustrated content, exercises at the end of chapters, and appendices that provide insights into advanced topics and current research areas, making it a valuable resource for mathematicians, physicists, and engineers.

This book is mainly intended as a textbook for students at the Sophomore-Junior level, majoring in mathematics, engineering, or the sciences in general. The book includes the basic topics in Ordinary Differential Equations, normally taught in an undergraduate class, as linear and nonlinear equations and systems, Bessel functions, Laplace transform, stability, etc. It is written with ample exibility to make it appropriate either as a course stressing applications, or a course stressing rigor and analytical thinking. This book also offers sufficient material for a one-semester graduate course, covering topics such as phase plane analysis, oscillation, Sturm-Liouville equations, Euler-Lagrange equations in Calculus of Variations, first and second order linear PDE in 2D. There are substantial lists of exercises at the ends of chapters. A solutions manual, containing complete and detailed solutions to all the exercises in the book, is available to instructors who adopt the book for teaching their classes.

Several important problems in Physics, Differential Geometry, and related fields lead to the consideration of semilinear variational elliptic equations on R, which have been the focus of extensive study. Mathematically, the primary interest lies in the limitations of Nonlinear Functional Analysis tools, particularly those based on compactness arguments, necessitating the development of new techniques. Additionally, many elliptic problems on R exhibit a perturbative nature. In certain cases, a natural perturbation parameter arises, such as in bifurcation from the essential spectrum, singularly perturbed equations, or the examination of semiclassical standing waves for NLS. In other instances, perturbations serve as a preliminary step toward achieving global results or offer a valuable perspective for further global studies. A specific approach that leverages this perturbative context appears most suitable. Perturbation methods in critical point theory provide the necessary abstract tools, proving applicable to a wide variety of equations typically regarded as distinct. This monograph aims to discuss these abstract methods alongside their applications to various perturbation problems, all of which share the common feature of involving semilinear Elliptic Partial Differential Equations on R with a variational structure.