On solutions with polynomial growth to an autonomous nonlinear elliptic problem

We study the following nonlinear elliptic problem [-Δu =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 $α\in {\mathbb R}^n$ and $C>0$ such that [(*) | u- α\cdot x | \leq C.] He also showed the existence of solutions with any prescribed $α\in {\mathbb R}^n$. In this note, we first prove that any solution satisfying (*) with nonzero vector $α$ 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 $|x|^d$.

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