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 $\mathbb R^N$.