The existence of partially localized periodic-quasiperiodic solutions and related KAM-type results for elliptic equations on the entire space

TL;DR

This paper proves the existence of numerous complex solutions to a class of elliptic equations on the entire space, exhibiting localized, periodic, and quasiperiodic behaviors, using advanced mathematical techniques.

Contribution

It introduces new KAM-type methods to establish the existence of partially localized, periodic, and quasiperiodic solutions for elliptic PDEs on unbounded domains.

Findings

Uncountably many positive solutions exist with specified symmetry and decay properties.

Solutions are periodic in one variable and quasiperiodic in another.

The methods extend to more general elliptic equations.

Abstract

We consider the equation $\Delta_x u+u_{yy}+f(u)=0,\ x=(x_1,\dots,x_N)\in\mathbb{R}^N,\ y\in \mathbb{R},$ where $N\geq 2$ and $f$ is a sufficiently smooth function satisfying $f(0)=0$, $f'(0)<0$, and some natural additional conditions. We prove that the equation possesses uncountably many positive solutions (disregarding translations) which are radially symmetric in $x'=(x_1,\dots,x_{N-1})$ and decaying as $|x'|\to\infty$, 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.