On solutions with polynomial growth to an autonomous nonlinear elliptic problem

Abstract

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 $\alpha \in {\mathbb{R}}^n$ and $C>0$ such that [(*) | u- α\cdot x | \leq C.] He also showed the existence of solutions with any prescribed $\alpha \in {\mathbb{R}}^n$. 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 $d\ge 1$ there exists a solution with polynomial growth $|x{|}^d$.

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