On solutions with polynomial growth to an autonomous nonlinear elliptic problem

We study the following nonlinear elliptic problem −∆u = F (u) in R where F (u) is a periodic function. Moser (1986) showed that for any minimal and nonself-intersecting solution, there exist α ∈ R and C > 0 such that (∗) |u− α · x| ≤ C. He also showed the existence of solutions with any prescribed α ∈ R. In this note, we first prove that any solution satisfying (*) with nonzero vector α must be one dimensional. Then we show that in R, for any positive integer d ≥ 1 there exists a solution with polynomial growth |x|. 1991 Mathematics Subject Classification. 35J25, 35B25, 35B23.