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

doi:10.1007/s10884-020-09925-5

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

School of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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, … , x_N_-₁) and decaying as | x^′| → ∞, periodic in x_N, 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 language	English (US)
Pages (from-to)	3035-3056
Number of pages	22
Journal	Journal of Dynamics and Differential Equations
Volume	34
Issue number	4
DOIs	https://doi.org/10.1007/s10884-020-09925-5
State	Accepted/In press - 2021

Bibliographical note

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

Publisher link

10.1007/s10884-020-09925-5

Cite this

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

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.",

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AU - Valdebenito, Darío A.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Center manifold

KW - Elliptic equations

KW - Entire solutions

KW - KAM theorems

KW - Partially localized solutions

KW - Quasiperiodic solutions

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

Abstract

Bibliographical note

Keywords

Publisher link

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