  We study the following nonlinear elliptic problem [-\Delta u =F^{'} (u) in
{\mathbb R}^n] where $F(u)$ is a periodic function. Moser (1986) showed that
for any minimal and nonself-intersecting solution, there exist $ \alpha \in
{\mathbb R}^n$ and $ C>0$ such that [(*) | u- \alpha \cdot x | \leq C.] He also
showed the existence of solutions with any prescribed $\alpha \in {\mathbb
R}^n$. In this note, we first prove that any solution satisfying (*) with
nonzero vector $\alpha$ must be one dimensional. Then we show that in ${\mathbb
R}^2$, for any positive integer $d\geq 1$ there exists a solution with
polynomial growth $|x|^d$.
