On solutions with polynomial growth to an autonomous nonlinear elliptic problem

TL;DR

This paper investigates solutions to a nonlinear elliptic equation with periodic nonlinearity, proving one-dimensionality for solutions with linear growth and constructing solutions with polynomial growth in two dimensions.

Contribution

It establishes that solutions with linear growth are one-dimensional and constructs solutions with polynomial growth in two dimensions, expanding understanding of solution behaviors.

Findings

Solutions with linear growth are one-dimensional.

Existence of solutions with polynomial growth in two dimensions.

Characterization of solutions based on growth rates.

Abstract

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 $|x|^d$.