Minimax theorems.

*(English)*Zbl 0856.49001
Progress in Nonlinear Differential Equations and their Applications. 24. Boston: Birkhäuser. viii, 159 p. (1996).

In this interesting book a variety of minimax type theorems and theorems on minimizing and Palais-Smale sequences are presented. Among them I mention the mountain pass theorem of Ambrosetti-Rabinowitz, the saddle-point theorem and linking theorem of Rabinovitz, Ekeland’s variational principle, quantitative deformation lemma of M. Willem, fountain theorem of Bartsch, and so on. They permit to prove the existence and multiplicity of the critical values of some functionals defined on Banach spaces and to characterize the critical points by means of the so-called Palais-Smale sequences, as well as to construct such sequences, …

Some generalization of classical Lyusternik-Shnirelman category theory (using relative category defined by Reeken) is also given. The developed theory is applied with success to nonlinear PDE’s in order to obtain the existence results; for instance, for semilinear Dirichlet problems (also with concave and convex nonlinearities), for semilinear Schrödinger equation (in this Section the Kryszewski-Szulkin degree theory is used), for generalized Kadomtsev-Petviashvili equation, where the existence of solitary waves is proved.

The material of the book covers many recent tools and some unpublished results in the domain of nonlinear PDE’s and their applications.

Some generalization of classical Lyusternik-Shnirelman category theory (using relative category defined by Reeken) is also given. The developed theory is applied with success to nonlinear PDE’s in order to obtain the existence results; for instance, for semilinear Dirichlet problems (also with concave and convex nonlinearities), for semilinear Schrödinger equation (in this Section the Kryszewski-Szulkin degree theory is used), for generalized Kadomtsev-Petviashvili equation, where the existence of solitary waves is proved.

The material of the book covers many recent tools and some unpublished results in the domain of nonlinear PDE’s and their applications.

Reviewer: Z.Denkowski (Kraków)

##### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

49J35 | Existence of solutions for minimax problems |

49K35 | Optimality conditions for minimax problems |