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

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Abstract

We consider the equation

$$\begin{aligned} \Delta _x u+u_{yy}+f(u)=0,\quad x=(x_1,\dots ,x_N)\in {{\mathbb {R}}}^N,\ y\in {{\mathbb {R}}}, \end{aligned}$$
(1)

where \(N\ge 2\) 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 \(x'=(x_1,\ldots ,x_{N-1})\) and decaying as \(|x'|\rightarrow \infty \), 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.

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Peter Poláčik

  2. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA

    Darío A. Valdebenito

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  1. Peter Poláčik
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  2. Darío A. Valdebenito
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Correspondence to Peter Poláčik.

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Dedicated to the memory of Geneviève Raugel

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Supported in part by the NSF Grant DMS–1856491.

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Poláčik, P., Valdebenito, D.A. The Existence of Partially Localized Periodic–Quasiperiodic Solutions and Related KAM-Type Results for Elliptic Equations on the Entire Space. J Dyn Diff Equat 34, 3035–3056 (2022). https://doi.org/10.1007/s10884-020-09925-5

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