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UBC Theses and Dissertations

Topics on critical point theory Fang, Guangcai


Many questions in mathematics and physics can be reduced to the problem of finding and classifying the critical points of a suitable functional on an appropriate manifold. In this thesis, we will be concerned with the problems of existence, location and structure of critical points by building upon the well known min-max methods that are presently used in non-linear differential equations. The thesis consists of two parts: In the first part, we exploit the new powerful mountain pass principle of Ghoussoub and Preiss and its higher dimensional extensions by Ghoussoub to classify the critical points generated by Min-Max methods. The functionals under study are only assumed to be C1and therefore the classical Morse theory is not available. In order to do this, we isolate various topological indices that can be associated with certain critical sets and points. If the functionals are C2 and the critical points are non-degenerated, these indices can then be used to recover the standard results on Morse indices. The study in this direction was first initiated in the case of the mountain pass theorem by Hofer and was expanded later by Pucci-Serrin and Ghoussoub-Preiss. We shall extend and simplify all the previously obtained structural results in this setting, but more importantly, we consider the case of the saddle point theorem and various other higher dimensional settings. In the last part, we construct an almost critical sequence (xn). by min- max procedures for a C2-functional co on a Hilbert space with some Morse type information. Actually we obtain some analytical (second order) properties concerning the Hessian d2co(x,i) which can be viewed as the asymptotic version of the information on the Morse index of the limit of(x„)„ whenever such a limit exists. As noted by P.L. Lions in his studies of the Hartree-Fock equations for Coulomb systems, this type of additional information about an almost critical sequence can sometimes be crucial in the proof of its convergence and in solving the corresponding variational problems. Examples are given as applications of the general theory developed in the thesis.

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