The Liouville property and a conjecture of De Giorgi

We consider bounded entire solutions of the nonlinear PDE Δu + u − u³ = 0 in ℝ^d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes. The proof uses a Liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator. © 2000 John Wiley & Sons, Inc.