An Introduction to Minimax Theorems and Their Applications to Differential Equations

This text is meant to be an introduction to critical point theory and its ap­ plications to differential equations. It is designed for graduate and postgrad­ uate students as well as for specialists in the fields of differential equations, variational methods and optimization. Although related material can be the treatment here has the following main purposes: found in other books, ¿ To present a survey on existing minimax theorems, ¿ To give applications to elliptic differential equations in bounded do­ mains and periodic second-order ordinary differential equations, ¿ To consider the dual variational method for problems with continuous and discontinuous nonlinearities, ¿ To present some elements of critical point theory for locally Lipschitz functionals and to give applications to fourth-order differential equa­ tions with discontinuous nonlinearities, ¿ To study homo clinic solutions of differential equations via the varia­ tional method. The Contents of the book consist of seven chapters, each one divided into several sections. A bibliography is attached to the end of each chapter. In Chapter I, we present minimization theorems and the mountain-pass theorem of Ambrosetti-Rabinowitz and some of its extensions. The con­ cept of differentiability of mappings in Banach spaces, the Fnkhet's and Gateaux derivatives, second-order derivatives and general minimization the­ orems, variational principles of Ekeland [EkI] and Borwein & Preiss [BP] are proved and relations to the minimization problem are given. Deformation lemmata, Palais-Smale conditions and mountain-pass theorems are consid­ ered.