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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