This article provides a survey on mountain pass theory. The mountain pass theory is a useful tool to find the critical points of functionals, and it plays a central role in analysis of nonlinear problems, especially nonlinear differential equations. Various versions of the mountain pass theorem have been investigated. We state the generalizations to nonsmooth functionals of three versions of the mountain pass theorem (the case of mountains of positive altitude), the Pucci-Serrin theorem (the case of mountains of zero altitude), and the Goussoub-Preiss theorem. Ekeland's variational principle is a central tool to prove these results on nonsmooth functionals. Finally, several relevant applications to semilinear elliptic partial differential equations are submitted.
The impact of the mountain pass theory in nonlinear analysis: a mathematical survey
PUCCI, Patrizia;
2010
Abstract
This article provides a survey on mountain pass theory. The mountain pass theory is a useful tool to find the critical points of functionals, and it plays a central role in analysis of nonlinear problems, especially nonlinear differential equations. Various versions of the mountain pass theorem have been investigated. We state the generalizations to nonsmooth functionals of three versions of the mountain pass theorem (the case of mountains of positive altitude), the Pucci-Serrin theorem (the case of mountains of zero altitude), and the Goussoub-Preiss theorem. Ekeland's variational principle is a central tool to prove these results on nonsmooth functionals. Finally, several relevant applications to semilinear elliptic partial differential equations are submitted.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
