The Existence of Partially Localized Periodic–Quasiperiodic Solutions and Related KAM-Type Results for Elliptic Equations on the Entire Space

Peter Poláčik, Darío A. Valdebenito

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the equation Δxu+uyy+f(u)=0,x=(x1,⋯,xN)∈RN,y∈R,where N≥ 2 and f is a sufficiently smooth function satisfying f(0) = 0 , f(0) < 0 , and some natural additional conditions. We prove that equation (1) possesses uncountably many positive solutions (disregarding translations) which are radially symmetric in x= (x1, … , xN-1) and decaying as | x| → ∞, periodic in xN, and quasiperiodic in y. Related theorems for more general equations are included in our analysis as well. Our method is based on center manifold and KAM-type results.

Original languageEnglish (US)
JournalJournal of Dynamics and Differential Equations
DOIs
StateAccepted/In press - 2021

Bibliographical note

Funding Information:
Supported in part by the NSF Grant DMS–1856491.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

Keywords

  • Center manifold
  • Elliptic equations
  • Entire solutions
  • KAM theorems
  • Partially localized solutions
  • Quasiperiodic solutions

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