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On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations
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## On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations

### Vasily Denisov and Andrey Muravnik

**Abstract.**
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.

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