Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity

Peter Poláčik, Darío A. Valdebenito

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1 Scopus citations

Abstract

We consider the equation ∆u + uyy + f(x, u) = 0, (x, y) ∈ RN × R (1) where f is sufficiently regular, radially symmetric in x, and f(·, 0) ≡ 0. We give sufficient conditions for the existence of solutions of (1) which are quasiperiodic in y and decaying as |x| → ∞ uniformly in y. Such solutions are found using a center manifold reduction and results from the KAM theory. A required nondegeneracy condition is stated in terms of fu(x, 0) and fuu(x, 0), and is independent of higher-order terms in the Taylor expansion of f(x, ·). In particular, our results apply to some quadratic nonlinearities.

Original languageEnglish (US)
Pages (from-to)1369-1393
Number of pages25
JournalDiscrete and Continuous Dynamical Systems - Series S
Volume13
Issue number4
DOIs
StatePublished - 2019

Bibliographical note

Funding Information:
37K55. Key words and phrases. Elliptic equations on the entire space, quasiperiodic solutions, center manifold, Birkhoff normal form, KAM theorem. The first author was supported in part by the NSF Grant DMS-1565388. The second author was supported in part by CONICYT-Chile Becas Chile, Convocatoria 2010. ∗ Corresponding author: P. Poláˇcik.

Publisher Copyright:
© 2019 American Institute of Mathematical Sciences. All rights reserved.

Keywords

  • Birkhoff normal form
  • Center manifold
  • Elliptic equations on the entire space
  • KAM theorem
  • Quasiperiodic solutions

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