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