By Antonio Ambrosetti (auth.), Michele Matzeu, Alfonso Vignoli (eds.)

Topological instruments in Nonlinear research had a massive advance­ ment over the past few a long time. the 3 major streams of study during this box, Topological measure, Singularity thought and Variational Meth­ ods, have in recent times develop into impetuous rivers of clinical research. the method continues to be occurring and the achievements during this region are fantastic. A such a lot promising and swiftly constructing box of analysis is the research of the position that symmetries play in nonlinear difficulties. Symmetries look in a rather usual manner in lots of difficulties in physics and in differential or symplectic geometry, similar to closed orbits for self reliant Hamiltonian platforms, configurations of symmetric elastic plates stressed, Hopf Bifurcation, Taylor vortices, convective motions of fluids, oscillations of chemical reactions, and so on . . . a few of these difficulties were tackled lately by way of assorted suggestions utilizing equivariant types of measure, Singularity and adaptations. the most objective of the current quantity is to offer a survey of a few of the main major achievements got by means of topological equipment in Nonlinear research over the last two-three a long time. The survey articles awarded the following replicate the private style and issues of view of the authors (all of them famous and individual experts of their personal fields) at the subject material. a typical function of those papers is that of begin­ ing with an old introductory history of different disciplines into consideration and mountain climbing as much as the heights of the newest re­ sults.

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We hope that our approach can overcome these difficulties and we hope that the Morse theory will get a bigger attention from people working in this field. We did not have enough time to write an other chapter on the periodic applications solutions of of Morse theory Hamiltonian to systems. the study This of work, started by Conley, Zahnder [Co,Ze,841 and, with a different approach by Ekeland [Ek,84] (cf. also the book [Ek,90]), is beautiful and full of interesting results. a recent paper of Dell' Antonio, We just mention D'Onofrio and Ekeland, 39 Introduction to Morse Theory [De,DO,Ek, tal, in which the Morse theory has been exploited to prove that the Hamiltonian systems, convex and even, are not ergodic in general.

Moreover, under a further assumption on the behaviour of V near x = 0 one can be more precise about the regularity of the solutions. When V is even the preceding result can be greatly sharpened. 7. Let V E C 2 (RN\{0}, R) satisfy the preceding assumptions. Moreover suppose V is even and there exists p > 0 such that (Vll) V(x) = -Ixl- o + 1jJ(lxl), vo < Ixl :::; p, 27 Variational Methods and Nonlinear Problems where 'ljJ satisfies sQ'ljJ(s) , SQ+1'ljJ'(s), sQ+2'ljJ"(s) -> 0 as s -> O. 7) has a non-sollision solution, provided that either N ~ 4 and 0 < a < 2, or N = 3 and 1 < a < 2.

12) + ~V'(x). x] = -00. 13) liminf [V(x) Ixl->+oo and lim [V(x) Ixl->o 2 2 Moreover one has that cat(Mh) = cat(A) = +00. 14) The advantage of working such a manifold relies on the fact that 90 satisfies the (PS) condition on M h . 4. 12) and (V6) 3a E (0,2) such that V'(x) . x 2: -aV(x), Vx i- 0; (V7) 3r > 0, (3 E (0,2) such that V'(x) . x::; -(3V(x), VO < Ixl < r. Then 90 satisfies (PS) on M h . Moreover, for all a > 0 the sublevels 9'8 are closed. 13). 14) allow us to use the Lusternik-Schnirelman theory.

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