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

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Luigi Ambrosio and Xavier Cabré
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## 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 $F$.

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## Additional Information

**Luigi Ambrosio**- Affiliation: Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
- MR Author ID: 25430
- Email: luigi@ambrosio.sns.it
**Xavier Cabré**- Affiliation: Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Diagonal, 647, 08028 Barcelona, Spain
- Email: cabre@ma1.upc.es
- Received by editor(s): October 8, 1999
- Published electronically: July 6, 2000
- Additional Notes: The authors would like to thank Mariano Giaquinta for several useful discussions. Most of this work was done while the second author was visiting the University of Pisa. He thanks the Department of Mathematics for its hospitality.
- © Copyright 2000 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**13**(2000), 725-739 - MSC (2000): Primary 35J60, 35B05, 35B40, 35B45
- DOI: https://doi.org/10.1090/S0894-0347-00-00345-3
- MathSciNet review: 1775735