Solutions of asymptotically linear operator equations via morse theory

✍ Scribed by Kung Ching Chang

• Publisher: John Wiley and Sons
• Year: 1981
• Tongue: English
• Weight: 648 KB
• Volume: 34
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160340503

✦ Synopsis

In a recent paper, H. Amann and E. Zehnder [ 11 studied existence problems for equations of the form (1)

in a real Hilbert space H. Here A is a selfadjoint linear operator and F is a potential operator, mapping H continuously into itself. It is well known that equation ( 1) is a good framework for studying the semilinear elliptic boundary value problems, the periodic solutions of a semilinear wave equation, and the periodic solutions of Hamiltonian systems. Suppose that F has a derivative at infinity, F'(oo), such that 0 @ Spect(A -F'( a))

and that the nonlinearity of F can only interact with finitely many eigenvalues of A. Under some additional mild conditions, which are satisfied in applications, they deduced the existence of at least one nontrivial solution of the equation ( 1).

The following theorem proved in their paper is crucial, and is achieved by means of a generalized Morse theory in the sense of C. C. Conley, and a big topological machinery.