Nonlinear Analysis and Semilinear Elliptic Problems
With many illustrations and helpful exercises, this book covers a number of more modern topics from critical point theory to elliptic partial differential equations.
TABLE OF CONTENTS Nonlinear Analysis and Semilinear Elliptic Problems
Part II: Variational Methods, I
Chapter List
- Chapter 5: Critical Points: Extrema
- Chapter 6: Constrained Critical Points
- Chapter 7: Deformations and the Palais Smale Condition
- Chapter 8: Saddle Points and Min-Max Methods
In this chapter we will discuss the existence of maxima and minima for a functional on a Hilbert or Banach space.
5.1 Functionals and Critical Points
Let E be a Banach space. A functional on E is a continuous real valued map J : E ? 
.
More in general, one could consider functionals defined on open subsets of E. But, for the sake of simplicity, in the sequel we will always deal with functionals defined on all of E, unless explicitly remarked.
Let J be (Fr chet) differentiable at u ? E with derivative d J(u) ? L(E, ). Recall that (see Section 1.1):
-
if J is differentiable on E, namely at every point u ? E and the map E
L(E, ), u d J(u), is continuous, we say that J ? C 1 (E, ); -
if J is k times differentiable on E with kth derivative d k J(u) ? L k (E,
) (the space of k-linear maps from E to ) and the application E L k (E, ), u d k J(u), is continuous, we say that J ? C k (E, ).
Definition...
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