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
Chapter 9: Lusternik Schnirelman Theory
In this chapter we discuss an elegant theory, introduced by Lusternik and Schnirelman, that allows us to find critical points of a functional J on a manifold M, in connection with the topological properties of M. In particular, this theory enables us to obtain multiplicity results.
General remark 9.1
In the sequel we will always understand that
9.1 The Lusternik Schnirelman Category
The main ingredient of the Lusternik Schnirelman (L-S, for short) theory is a topological tool, the L-S category, that we are going to define.
Let M be a topological space. A subset A of M is contractible in M if the inclusion i : A ? M is homotopic to a constant p ? M, namely if there exists H ? C([0, 1] A, M) such that H(0, u) = u and H(1, u) = p.
Definition 9.2
The (L-S) category of A with respect to M (or simply the category of A with respect to M), denoted by cat (A , M), is the least integer k such that A ? A 1 ? ? A k , with A i (i = 1, , k) closed and contractible in M. We set cat ( 
, M) = 0 and cat (A, M) = + ? if there are no integers with the above property. We will use...
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