Entire solutions of semilinear elliptic equations in ℝ³ and a conjecture of De Giorgi

Abstract

In 1978 De Giorgi formulated the following conjecture. Let u u be a solution of Δ u = u 3 − u \Delta u=u^{3}-u in all of R n \mathbb {R}^{n} such that | u | ≤ 1 \vert u\vert \le 1 and ∂ n u > 0 \partial _{n} u >0 in R n \mathbb {R}^{n} . Is it true that all level sets { u = λ } \{ u=\lambda \} of u u are hyperplanes, at least if n ≤ 8 n\le 8\, ? Equivalently, does u u depend only on one variable? When n = 2 n=2 , this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for n = 3 n=3 . The question, however, remains open for n ≥ 4 n\ge 4 . The results for n = 2 n=2 and 3 apply also to the equation Δ u = F ′ ( u ) \Delta u=F’(u) for a large class of nonlinearities  F F .