Entire solutions of semilinear elliptic equations in <IMG > and a conjecture of De Giorgi期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Luigi Ambrosio Xavier Cabré

Institution:	Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126 Pisa, Italy ; Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Diagonal, 647, 08028 Barcelona, Spain

Abstract:	In 1978 De Giorgi formulated the following conjecture. Let 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 are hyperplanes, at least if $n\le 8\,$ ? 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 $n\ge 4$ . The results for and 3 apply also to the equation $\Delta u=F'(u)$ for a large class of nonlinearities .

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

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