We consider the equation \begin{equation}\tag{1} \Delta_x u+u_{yy}+f(u)=0,\quad x=(x_1,\dots,x_N)\in\mathbb{R}^{N},\ y\in \mathbb{R}, \end{equation} 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 equation (1) 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.