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

Authors:
Luigi Ambrosio and Xavier Cabré

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

Published electronically:
July 6, 2000

MathSciNet review:
1775735

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Abstract: In 1978 De Giorgi formulated the following conjecture. *Let ** be a solution of * * in all of * *such that * * and * * in * *. Is it true that all level sets * * of ** are hyperplanes, at least if *? Equivalently, does depend only on one variable? When , this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for . The question, however, remains open for . The results for and 3 apply also to the equation for a large class of nonlinearities .

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Additional Information

**Luigi Ambrosio**

Affiliation:
Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126 Pisa, Italy

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

DOI:
https://doi.org/10.1090/S0894-0347-00-00345-3

Keywords:
Nonlinear elliptic PDE,
symmetry and monotonicity properties,
energy estimates,
Liouville theorems

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.

Article copyright:
© Copyright 2000
American Mathematical Society