Entire solutions of semilinear elliptic equations in $\mathbb {R}^{3}$ and a conjecture of De Giorgi

Abstract:

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