Skip to main content
Top

2018 | OriginalPaper | Chapter

Four Conjectures in Nonlinear Analysis

Author : Biagio Ricceri

Published in: Applications of Nonlinear Analysis

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this chapter, I formulate four challenging conjectures in Nonlinear Analysis. More precisely: a conjecture on the Monge-Ampère equation; a conjecture on an eigenvalue problem; a conjecture on a non-local problem; a conjecture on disconnectedness versus infinitely many solutions.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference A. Bahri, J.M. Coron, Sur une équation elliptique non linéaire avec l’exposant critique de Sobolev. C. R. Acad. Sci. Paris Sér. I Math. 301, 345–348 (1985) A. Bahri, J.M. Coron, Sur une équation elliptique non linéaire avec l’exposant critique de Sobolev. C. R. Acad. Sci. Paris Sér. I Math. 301, 345–348 (1985)
2.
go back to reference A. Bahri, J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Commun. Pure Appl. Math. 41, 253–294 (1988)MathSciNetCrossRef A. Bahri, J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Commun. Pure Appl. Math. 41, 253–294 (1988)MathSciNetCrossRef
3.
go back to reference M. Clapp, F. Pacella, Multiple solutions to the pure critical exponent in domains with a hole of arbitrary size. Math. Z. 259, 575–589 (2008) M. Clapp, F. Pacella, Multiple solutions to the pure critical exponent in domains with a hole of arbitrary size. Math. Z. 259, 575–589 (2008)
4.
go back to reference J.M. Coron, Topologie et cas limite des injections de Sobolev. C. R. Acad. Sci. Paris Sér. I Math. 299, 209–212 (1984) J.M. Coron, Topologie et cas limite des injections de Sobolev. C. R. Acad. Sci. Paris Sér. I Math. 299, 209–212 (1984)
6.
go back to reference Z. Denkowski, S. Migórski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory (Kluwer Academic, Boston, 2003)CrossRef Z. Denkowski, S. Migórski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory (Kluwer Academic, Boston, 2003)CrossRef
7.
go back to reference L. Diening, C. Kreuzer, S. Schwarzacher, Convex hull property and maximum principle for finite element minimisers of general convex functionals. Numer. Math. 124, 685–700 (2013)MathSciNetCrossRef L. Diening, C. Kreuzer, S. Schwarzacher, Convex hull property and maximum principle for finite element minimisers of general convex functionals. Numer. Math. 124, 685–700 (2013)MathSciNetCrossRef
8.
go back to reference R. Engelking, Theory of Dimensions, Finite and Infinite (Heldermann, Lemgo, 1995) R. Engelking, Theory of Dimensions, Finite and Infinite (Heldermann, Lemgo, 1995)
9.
go back to reference X.L. Fan, A remark on Ricceri’s conjecture for a class of nonlinear eigenvalue problems. J. Math. Anal. Appl. 349, 436–442 (2009)MathSciNetCrossRef X.L. Fan, A remark on Ricceri’s conjecture for a class of nonlinear eigenvalue problems. J. Math. Anal. Appl. 349, 436–442 (2009)MathSciNetCrossRef
10.
go back to reference X.L. Fan, On Ricceri’s conjecture for a class of nonlinear eigenvalue problems. Appl. Math. Lett. 22, 1386–1389 (2009)MathSciNetCrossRef X.L. Fan, On Ricceri’s conjecture for a class of nonlinear eigenvalue problems. Appl. Math. Lett. 22, 1386–1389 (2009)MathSciNetCrossRef
11.
go back to reference X.L. Fan, B. Ricceri, On the Dirichlet problem involving nonlinearities with non-positive primitive: a problem and a remark. Appl. Anal. 89, 189–192 (2010)MathSciNetCrossRef X.L. Fan, B. Ricceri, On the Dirichlet problem involving nonlinearities with non-positive primitive: a problem and a remark. Appl. Anal. 89, 189–192 (2010)MathSciNetCrossRef
12.
go back to reference N. Hirano, Existence of nontrivial solutions for a semilinear elliptic problem with supercritical exponent. Nonlinear Anal. 55, 543–556 (2003)MathSciNetCrossRef N. Hirano, Existence of nontrivial solutions for a semilinear elliptic problem with supercritical exponent. Nonlinear Anal. 55, 543–556 (2003)MathSciNetCrossRef
13.
go back to reference N.I. Katzourakis, Maximum principles for vectorial approximate minimizers of nonconvex functionals. Calc. Var. Partial Differ. Equ. 46, 505–522 (2013)MathSciNetCrossRef N.I. Katzourakis, Maximum principles for vectorial approximate minimizers of nonconvex functionals. Calc. Var. Partial Differ. Equ. 46, 505–522 (2013)MathSciNetCrossRef
14.
go back to reference J.L. Kazdan, F.W. Warner, Remarks on some quasilinear elliptic equations. Commun. Pure Appl. Math. 28, 567–597 (1975)MathSciNetCrossRef J.L. Kazdan, F.W. Warner, Remarks on some quasilinear elliptic equations. Commun. Pure Appl. Math. 28, 567–597 (1975)MathSciNetCrossRef
15.
go back to reference J.L. Kelley, I. Namioka, Linear Topological Spaces (Van Nostrand, Princeton, 1963)CrossRef J.L. Kelley, I. Namioka, Linear Topological Spaces (Van Nostrand, Princeton, 1963)CrossRef
16.
go back to reference A.J.B. Lopes-Pinto, On a new result on the existence of zeros due to Ricceri. J. Convex Anal. 5, 57–62 (1998)MathSciNetMATH A.J.B. Lopes-Pinto, On a new result on the existence of zeros due to Ricceri. J. Convex Anal. 5, 57–62 (1998)MathSciNetMATH
17.
go back to reference D. Passaseo, Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains. Manuscripta Math. 65, 147–175 (1989)MathSciNetCrossRef D. Passaseo, Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains. Manuscripta Math. 65, 147–175 (1989)MathSciNetCrossRef
18.
go back to reference D. Passaseo, Nontrivial solutions of elliptic equations with supercritical exponent in contractible domains. Duke Math. J. 92, 429–457 (1998)MathSciNetCrossRef D. Passaseo, Nontrivial solutions of elliptic equations with supercritical exponent in contractible domains. Duke Math. J. 92, 429–457 (1998)MathSciNetCrossRef
19.
go back to reference S.I. Pohozaev, Eigenfunctions of the equation Δu + λf(u) = 0. Soviet Math. Dokl., 6, 1408–1411 (1965) S.I. Pohozaev, Eigenfunctions of the equation Δu + λf(u) = 0. Soviet Math. Dokl., 6, 1408–1411 (1965)
21.
go back to reference B. Ricceri, Applications of a theorem concerning sets with connected sections. Topol. Methods Nonlinear Anal. 5, 237–248 (1995)MathSciNetCrossRef B. Ricceri, Applications of a theorem concerning sets with connected sections. Topol. Methods Nonlinear Anal. 5, 237–248 (1995)MathSciNetCrossRef
22.
23.
24.
25.
go back to reference B. Ricceri, Energy functionals of Kirchhoff-type problems having multiple global minima. Nonlinear Anal. 115, 130–136 (2015)MathSciNetCrossRef B. Ricceri, Energy functionals of Kirchhoff-type problems having multiple global minima. Nonlinear Anal. 115, 130–136 (2015)MathSciNetCrossRef
26.
28.
29.
go back to reference E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. III (Springer, New York, 1985)CrossRef E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. III (Springer, New York, 1985)CrossRef
30.
go back to reference E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. I (Springer, New York, 1986)CrossRef E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. I (Springer, New York, 1986)CrossRef
Metadata
Title
Four Conjectures in Nonlinear Analysis
Author
Biagio Ricceri
Copyright Year
2018
DOI
https://doi.org/10.1007/978-3-319-89815-5_24

Premium Partner