By Kung-ching Chang (auth.)

The e-book relies on my lecture notes "Infinite dimensional Morse conception and its applications", 1985, Montreal, and one semester of graduate lectures brought on the college of Wisconsin, Madison, 1987. because the goal of this monograph is to provide a unified account of the subjects in serious aspect concept, a large amount of new fabrics has been extra. a few of them have by no means been released formerly. The e-book is of curiosity either to researchers following the advance of latest effects, and to humans looking an creation into this concept. the most effects are designed to be as self-contained as attainable. And for the reader's comfort, a few initial heritage details has been equipped. the next humans deserve designated thank you for his or her direct roles in support ing to organize this publication. Prof. L. Nirenberg, who first brought me to this box ten years in the past, whilst I visited the Courant Institute of Math Sciences. Prof. A. Granas, who invited me to provide a chain of lectures at SMS, 1983, Montreal, after which the above notes, because the basic model of part of the manuscript, that have been released within the SMS assortment. Prof. P. Rabinowitz, who supplied a lot wanted encouragement throughout the educational semester, and invited me to coach a semester graduate path and then the lecture notes grew to become the second one model of components of this e-book. Professors A. Bahri and H. Brezis who advised the book of the e-book within the Birkhiiuser series.

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**Additional resources for Infinite Dimensional Morse Theory and Multiple Solution Problems**

**Example text**

Suppose fECI (M, ~ 1 ). Let N' c N be two closed neighborhoods satisfying dist(N',8N) 2: ~8, 8> o. Suppose that there are constants band f: positive, such that IWex)1I 2: b 'v' x E fc+< \(fc-< UN'), 0< f < Min {~8b2, ~8b}. 3. 3. According to (PS)c, Kc is compact. Hence, for 8> 0 sufficiently small, N(8) = {x E Mldist(x,Kc) < 8} eN. 4, if we take N' = N(~). 4. Define a smooth function: pes) = { ~ for s rt. [e - f:, c + f:J, for s E [e - f, C + f], with 0::; pes) ::; 1. Let A = M\(N'h, where (N')o = {x E Mldist(x, N') ::; 8 8}, and B = N' be two closed subsets.

E. Combining T/ with ~, we obtain the deformation retract. 3) is easily verified. This completes the proof. 4 are Milnor [Mill], Schwartz [ScJl], Rothe [Rotl], Palais [Pall], Pitcher [Pitl] and Marino Prodi [MaPl]. 4, the handle body theorem is established on Hilbert Riemannian manifolds, where the Morse Lemma holds, and the local behavior of a nondegenerate critical point is quite clear. In order to extend this theorem to Finsler manifolds, or to Banach spaces, new difficulties arise in two ways.

In fact (1), (3), and (4) follow directly from the construction. (2) holds if IIbll is small. Furthermore, 9 differs from f only in the neighborhood cp-l B(9, 6). Since Ho n B(9,6) is compact (dimHo = dimker~ f(p) < +00) and d 2 g 0 cp-l I(H+ffiH_)nB(8,6) is invertible, we conclude that 9 satisfies the (PS) condition if f does. 6. 1, in which we assumed the negative gradient points inward at the boundary. The purpose of this section is to extend the previous study in two aspects: (1) under more general boundary conditions, and (2) when the underlying space is not a manifold but a locally convex closed set.