New entire solutions to some classical semilinear elliptic problems

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Citations (Scopus)

Abstract

This paper deals with the construction of solutions to autonomous semilinear elliptic equations considered in entire space. In the absence of space dependence or explicit geometries of the ambient space, the point is to unveil internal mechanisms of the equation that trigger the presence of families of solutions with interesting concentration patterns. We discuss the connection between minimal surface theory and entire solutions of the Allen-Cahn equation. In particular, for dimensions 9 or higher, we build an example that provides a negative answer to a celebrated question by De Giorgi for this problem. We will also discuss related results for the (actually more delicate) standing wave problem in nonlinear Schrödinger equations and for sign-changing solutions of the Yamabe equation.

Original languageEnglish
Title of host publicationProceedings of the International Congress of Mathematicians 2010, ICM 2010 (4 vol set)
EditorsR. Bhatia, A. Pal, G. Rangarajan, V. Srinivas, M. Vanninathan
PublisherWorld Scientific Publishing
Pages1934-1957
Number of pages24
ISBN (Print)9789814324304
Publication statusPublished - 6 Aug 2011
EventInternational Congress of Mathematicians 2010, ICM 2010 - Hyderabad, India
Duration: 19 Aug 201027 Aug 2010

Publication series

NameProceedings of the International Congress of Mathematicians 2010, ICM 2010

Conference

ConferenceInternational Congress of Mathematicians 2010, ICM 2010
CountryIndia
CityHyderabad
Period19/08/1027/08/10

Keywords

  • Allen-Cahn equation
  • Standing waves for NLS
  • Yamabe equation

ASJC Scopus subject areas

  • Mathematics(all)

Fingerprint Dive into the research topics of 'New entire solutions to some classical semilinear elliptic problems'. Together they form a unique fingerprint.

Cite this