zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A general variational principle and some of its applications. (English) Zbl 0946.49001
The main result of the paper is the following variational principle type theorem: Let X be a topological space, and let Φ,Ψ:XR be two sequentially l.s.c. functions. Denote by I the set of all ρ>inf X Ψ such that the set Ψ -1 (]-,ρ[) is contained in some sequentially compact subset of X· Assume that I· For each ρI denote by ρ the family of all sequentially compact subsets of X containing Ψ -1 (]-,ρ[), and put α(ρ)=sup K ρ inf K Φ· Then, for each ρI and each λ satisfying λ>inf xΨ -1 (]-,ρ[) (Φ(x)-α(ρ))/(ρ-Ψ(x)), the restriction of the function Φ+λΨ to Ψ -1 (]-,ρ[) has a global minimum. Among several applications it is provided an existence result for nonlinear elliptic equations.

49J27Optimal control problems in abstract spaces (existence)
49J45Optimal control problems involving semicontinuity and convergence; relaxation
35J65Nonlinear boundary value problems for linear elliptic equations
47J30Variational methods (nonlinear operator equations)