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We study the Dirichlet problem in half-space for the equation $\nobreak{\Delta u+g(u)|\nabla u|^2=0,}$ where $g$ is continuous or has a power singularity (in the~latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as $y\to\infty,$ where $y$ denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function.

Copyright 2003 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 09 (2003), pp. 88-93
• Publisher Identifier: S 1079-6762(03)00115-X
• 2000 Mathematics Subject Classification. Primary 35J25; Secondary 35B40, 35J60
• Key words and phrases. Asymptotic behaviour of solutions, BKPZ-type non-linearities
• Received by editors March 6, 2002
• Posted on September 29, 2003
• Communicated by Michael E. Taylor
• Comments (When Available)

Vasily Denisov

Moscow State University, Faculty of Computational Mathematics and Cybernetics, Leninskie gory, Moscow 119899, Russia

E-mail address: V.Denisov@g23.relcom.ru

Andrey Muravnik

Department of Differential Equations, Moscow State Aviation Institute, Volokolamskoe shosse 4, Moscow, A-80, GSP-3, 125993, Russia

E-mail address: abm@mailru.com

The second author was supported by INTAS, grant 00-136 and RFBR, grant 02-01-00312.