Entire Solutions of Semilinear Elliptic Equations in R3 and a Conjecture of De Giorgi
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چکیده
In 1978 De Giorgi formulated the following conjecture. Let u be a solution of u = u 3 ? u in all of R n such that juj 1 and @ n u > 0 in R n. Is it true that all level sets fu = g of u are hyperplanes, at least if n 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 4. The results for n = 2 and 3 apply also to the equation u = F 0 (u) for a large class of nonlinearities F .
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تاریخ انتشار 2000