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 \[ u\in H^ 1(S^ 3),\quad u>0\quad in\quad S^ 3,\quad -8\Delta u+6u=Ku^ 5\quad in\quad S^ 3. \]

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 |