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An upper bound for positive solutions of the equation $\Delta u=u^\alpha$
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## An upper bound for positive solutions of the equation $\Delta u=u^\alpha$

### S. E. Kuznetsov

**Abstract.**
In 2002 Mselati proved that every positive solution of the
equation $\Delta u=u^2$ in a bounded domain of class $C^4$ is the
limit of an increasing sequence of moderate solutions. (A solution
is called moderate if it is dominated by a harmonic function.) As
a part of his proof, he established an upper bound (in terms of the
capacity of $K$) for solutions vanishing off a compact subset $K$
of $\partial E$. We use a different kind of capacity (we call it the
Poisson capacity) and we establish in terms of this capacity an
upper bound for solutions of $\Delta u=u^\alpha$ with $1<\alpha\le
2$. This is a part of the program: to classify all positive
solutions of this equation.

*Copyright 2004 American Mathematical Society
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#### Article Info

- ERA Amer. Math. Soc.
**10** (2004), pp. 103-112
- Publisher Identifier: S 1079-6762(04)00135-0
- 2000
*Mathematics Subject Classification*. Primary 35J15; Secondary 35J25
- Received by editors April 5, 2004
- Posted on September 27, 2004
- Communicated by Mark Freidlin
- Comments (When Available)

**S. E. Kuznetsov**

Department of Mathematics, University of Colorado, Boulder, CO
80309-0395

*E-mail address:* `Sergei.Kuznetsov@Colorado.edu`

Partially supported by the National Science Foundation Grant DMS-9971009

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