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

Peter Poláčik; Darío A. Valdebenito

doi:10.3934/dcdss.2020077

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2020, Volume 13, Issue 4: 1369-1393. Doi: 10.3934/dcdss.2020077

This issue Previous Article Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization Next Article Effective Hamiltonian dynamics via the Maupertuis principle

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

Peter Poláčik^1, , and
Darío A. Valdebenito^2,

1.
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States
2.
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada

^* Corresponding author: P. Poláčik
^* Corresponding author: P. Poláčik

Received: August 31, 2017

Revised: March 31, 2018

Published: March 2020

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

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Abstract

We consider the equation

$\Delta u+{{u}_{yy}}+f(x,u) = 0,\quad (x,y)\in {{\mathbb{R}}^{N}}\times \mathbb{R}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$

where $ f $ is sufficiently regular, radially symmetric in $ x $, and $ f(\cdot,0)\equiv 0 $. We give sufficient conditions for the existence of solutions of (1) which are quasiperiodic in $ y $ and decaying as $ |x|\to\infty $ 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 $ f_u(x,0) $ and $ f_{uu}(x,0) $, and is independent of higher-order terms in the Taylor expansion of $ f(x,\cdot) $. In particular, our results apply to some quadratic nonlinearities.

Keywords:

Mathematics Subject Classification: Primary: 35B08, 35B15, 35J61; Secondary: 37J40, 37K55.

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