We consider elliptic equations on $\mathbb{R}^{N+1}$ of the form
\[\tag{1}
\Delta_x u+u_{yy}+g(x,u)=0,\quad(x,y)\in \mathbb{R}^{N}\times\mathbb{R},
\]where $g(x,u)$ is a sufficiently regular function with $g(\cdot,0)\equiv 0$. We give sufficient conditions for the existence of solutions of (1) which are quasiperiodic in $y$ and decaying as $|x|\to\infty$ uniformly in $y$. Such solutions are found using a center manifold reduction and results from the KAM theory. We discuss several classes of nonlinearities $g$ to which our results apply.