On solutions with polynomial growth to an autonomous nonlinear elliptic problem

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 ${\mathbb R}^2$,foranypositiveinteger$d\geq 1$ there exists a solution with polynomial growth $\mathrm{\mid }x{\mathrm{\mid }}^d$.