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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 $f$ isa sufficiently smooth function satisfying $f(0)=0$, $f'(0)<0$, and some naturaladditional conditions. We prove that the equation possesses uncountably manypositive 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 areincluded in our analysis as well. Our method is based on center manifold andKAM-type results.

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