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 critical point theorem via the Ekeland variational principle. (English) Zbl 1239.58011
Summary: The aim of this paper is to establish the existence of a local minimum for a continuously Gâteaux differentiable function, possibly unbounded from below, without requiring any weak continuity assumption. Several special cases are also emphasized. Moreover, a novel definition of Palais-Smale condition, which is more general than the usual one, is presented and a mountain pass theorem is pointed out. As a consequence, multiple critical points theorems are then established. Finally, as an example of applications, an elliptic Dirichlet problem with critical exponent is investigated.

58E30Variational principles on infinite-dimensional spaces
49J52Nonsmooth analysis (other weak concepts of optimality)
49J50Fréchet and Gateaux differentiability
35J40Higher order elliptic equations, boundary value problems
46E05Lattices of continuous, differentiable or analytic functions
58C20Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
[1]Ricceri, B.: On a three critical points theorem, Arch. math. (Basel) 75, 220-226 (2000) · Zbl 0979.35040 · doi:10.1007/s000130050496
[2]Ricceri, B.: A three critical points theorem revisited, Nonlinear anal. 70, 3084-3089 (2009) · Zbl 1214.47079 · doi:10.1016/j.na.2008.04.010
[3]Ricceri, B.: A further three critical points theorem, Nonlinear anal. 71, 4151-4157 (2009) · Zbl 1187.47057 · doi:10.1016/j.na.2009.02.074
[4]Averna, D.; Bonanno, G.: A three critical points theorem and its applications to the ordinary Dirichlet problem, Topol. methods nonlinear anal. 22, 93-103 (2003) · Zbl 1048.58005
[5]Bonanno, G.: Some remarks on a three critical points theorem, Nonlinear anal. 54, 651-665 (2003) · Zbl 1031.49006 · doi:10.1016/S0362-546X(03)00092-0
[6]Bonanno, G.; Candito, P.: Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. differential equations 244, 3031-3059 (2008) · Zbl 1149.49007 · doi:10.1016/j.jde.2008.02.025
[7]Ghoussoub, N.: Duality and perturbation methods in critical ppoint theory, Cambridge tracts in mathematics 107 (1993) · Zbl 0790.58002
[8]Motreanu, D.; Rădulescu, V.: Variational and non-variational methods in nonlinear analysis and boundary value problems, Nonconvex optimization and applications (2003)
[9]Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations, CBMS reg. Conferences in math. 65 (1985) · Zbl 0609.58002
[10]Zeidler, E.: Nonlinear functional analysis and its applications, Nonlinear functional analysis and its applications (1985) · Zbl 0583.47051
[11]Carl, S.; Dietrich, H.: The weak upper and lower solution method for quasilinear elliptic equations with generalized subdifferentiable perturbations, Appl. anal. 56, 263-278 (1995) · Zbl 0832.35039 · doi:10.1080/00036819508840326
[12]Costa, D. G.; Goncalves, J. V. A.: Critical point theory for nondifferentiable functionals and applications, J. math. Anal. appl. 153, 470-485 (1990) · Zbl 0717.49007 · doi:10.1016/0022-247X(90)90226-6
[13]Ricceri, B.: A general variational principle and some of its applications, J. comput. Appl. math. 113, 401-410 (2000) · Zbl 0946.49001 · doi:10.1016/S0377-0427(99)00269-1
[14]Bonanno, G.; D’aguì, G.: A critical point theorem and existence results for a nonlinear boundary value problem, Nonlinear anal. 72, 1977-1982 (2010) · Zbl 1200.34020 · doi:10.1016/j.na.2009.09.039
[15]Livrea, R.; Marano, S. A.: Existence and classification of critical points for nondifferentiable functions, Adv. differential equations 9, 961-978 (2004) · Zbl 1100.58008
[16]Pucci, P.; Serrin, J.: A mountain pass theorem, J. differential equations 63, 142-149 (1985) · Zbl 0585.58006 · doi:10.1016/0022-0396(85)90125-1
[17]Bonanno, G.; Bisci, G. Molica: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. value probl. 2009, 1-20 (2009) · Zbl 1177.34038 · doi:10.1155/2009/670675
[18]Bonanno, G.; Marano, S. A.: On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. anal. 89, 1-10 (2010) · Zbl 1194.58008 · doi:10.1080/00036810903397438
[19]Brézis, H.: Analyse fonctionelle–théorie et applications, (1983) · Zbl 0511.46001
[20]Talenti, G.: Best constants in Sobolev inequality, Ann. mat. Pura appl. 110, 353-372 (1976) · Zbl 0353.46018 · doi:10.1007/BF02418013
[21]Brézis, H.; Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. pure appl. Math. 36, 437-477 (1983) · Zbl 0541.35029 · doi:10.1002/cpa.3160360405
[22]Willem, M.: Minimax theorems, (1996)