On a notion of category depending on a functional. I: Theory and application to critical point theory. (English) Zbl 1185.58006
A notion of category for a functional \(f\) defined on a topological space is introduced. This notion permits us to obtain a better lower bound to the number of critical points of \(f\) than the bound obtained with the Ljusternik-Schnirelman category. Roughly speaking, the authors cover a set by contractible sets with suitable deformations \(\eta\). They also introduce and study the notions of relative category and truncated category depending on the functional \(f\). Next, it is shown that these notions of category depending on \(f\) permit to obtain a lower bound to the number of critical points of \(f\). Finally, the notion of limit relative category depending on the functional \(f\) is introduced and its properties are studied. An application to the existence and the multiplicity of solutions to Hamiltonian systems is also provided in the present paper.
Reviewer: Vicenţiu D. Rădulescu (Craiova)
MSC:
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |
Keywords:
Lusternik-Schnirelman category; critical point theory; deformation map associated to a functionalReferences:
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