We consider the equation
\begin{equation}
\Delta u+u_{yy}+f(u)=0,\quad (x,y)\in\mathbb R^N\times\mathbb R, \qquad (1)
\end{equation}
where $N\geq 2$ and $f$ is a smooth function satisfying $f(0)=0$ and
$f'(0)<0$. We show that for suitable nonlinearities $f$ of this
form equation (1) possesses uncountably many positive solutions
which are quasiperiodic in $y$, radially symmetric in $x$, and
decaying as $|x|\to\infty$ uniformly in $y$. Our method is based on
center manifold and KAM-type results and involves analysis of
solutions of (1) in a vicinity of a $y$-independent solution
$u^*(x)$--a ground state of the equation $\Delta u+f(u)=0$ on
$\mathbb R^N$.