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Critical points at infinity in some variational problems. (English) Zbl 0676.58021
Pitman Research Notes in Mathematics Series, 182. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons. v, 307 p. £22.00 (1989).
The book is devoted to the study of some variational problems in differential geometry and nonlinear analysis (Yamabe problem and Kazdan- Warner problem), in which the condition of Palais-Smale is violated. The solutions of such problems can be found as critical points of a functional f on a manifold $$\Sigma$$. A crucial point is that it is possible to define some subsets V(p,$$\epsilon)$$ of $$\Sigma$$ such that every Palais-Smale sequence $$(u_ k)$$ weakly converging to u satisfies the inclusion $u_ k-u/\| u_ k-u\| \in V(p,\epsilon_ k)$ for some fixed p and some $$\epsilon_ k\to 0.$$
The first part of the book is then devoted to a careful analysis of the behavior of the flow-lines of the gradient flow on the sets V(p,$$\epsilon)$$.
In the second part the informations on the sets V(p,$$\epsilon)$$ are related with deformation techniques and topological arguments, in order to prove the existence of a critical point of f on $$\Sigma$$, in spite of the fact that Palais-Smale condition does not hold.
About the Yamabe problem in an open proper subset of $$S^ n$$ (which is reduced, by stereographic projection, to a problem in $${\mathbb{R}}^ n)$$, the following result is proved: Theorem. Let $$\Omega$$ be a connected bounded open regular subset of $${\mathbb{R}}^ n$$ (n$$\geq 3)$$ such that $$H_ q(\Omega;{\mathbb{Z}}_ 2)\neq 0$$ for some $$q\geq 1$$. Then there exists a solution u to the problem $u\in H^ 1_ 0(\Omega),\quad u>0\quad in\quad \Omega,\quad -\Delta u=u^ p\quad in\quad \Omega$ where $$p=(n+2)/(n-2).$$
About the Kazdan-Warner problem the following result is proved: Theorem. Let K: $$S^ 3\to {\mathbb{R}}$$ be a strictly positive function of class $$C^ 2$$ with non-degenerate critical points $$y_ 1,...,y_ m$$ of Morse index $$k_ 1,...,k_ m$$. Let us assume that $$-8\Delta K(y_ i)+6K(y_ i)\neq 0$$ for every i and let $$I=\{i:\quad -8\Delta K(y_ i)+6K(y_ i)>0\}.$$ If $$\sum_{i\in I}(-1)^{k_ i}\neq -1,$$ then there exists a solution u to the problem $u\in H^ 1(S^ 3),\quad u>0\quad in\quad S^ 3,\quad -8\Delta u+6u=Ku^ 5\quad in\quad S^ 3.$
Reviewer: M.Degiovanni

##### MSC:
 58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals 58-02 Research exposition (monographs, survey articles) pertaining to global analysis 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35J65 Nonlinear boundary value problems for linear elliptic equations 58E30 Variational principles in infinite-dimensional spaces