In this course, our main aim is to develop abstract variational techniques which can be employed to study the existence of solutions of various Semi-linear elliptic Partial Differential Equations. The main fundamental results, that will be covered in this course, are functional analytic in nature and can be used in many other situations. A basic outline of the course is as follows:
The Pohozaev identity and non-existence of solutions.
Calculus in normed linear space: Fréchet and Gâteaux differentiability, Notion of integral, Existence and uniqueness theorem for ODE in Banach space.
Dirichlet’s principle, Basics of Sobolev spaces, Connection between critical points and solutions of PDE
Direct Methods in Calculus of Variations:Existence of extrema, Ekeland’s Variational Principle, Constrained critical points (method of Lagrange Multiplier).
Deformation and the Palais-Smale condition, Saddle points and min-max methods: The mountain pass theorem and its
application, The concentration compactness lemmas and their application.
Additional topics (to be covered if time permits): Linking Theorem,Index Theory, The Brezis-Nirenberg Problem.
Suggested books and references:
Ambrosetti, A, and Malchiodi, A, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2007.
Struwe, Michael, Variational methods, Applications to nonlinear partial differential equations and Hamiltonian systems, Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 34. Springer-Verlag, Berlin 1996.