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Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation $L u=u^\alpha$ with $\alpha>1$ in a bounded smooth domain $E\subset \mathbb{R}^d$. In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called $\sigma$-moderate. The question if all solutions belong to this class remained open. In 2002 Mselati proved that this is true for the equation $\Delta u=u^2$ in a domain of class $C^4$. His principal tool---the Brownian snake---is not applicable to the case $\alpha\neq 2$. In 2003 Dynkin and Kuznetsov modified most of Mselati's arguments by using superdiffusions instead of the snake. However a critical gap remained. A new inequality established in the present paper allows us to close this gap.

Copyright 2004 American Mathematical Society
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Article Info

• ERA Amer. Math. Soc. 10 (2004), pp. 68-77
• Publisher Identifier: S 1079-6762(04)00131-3
• 2000 Mathematics Subject Classification. Primary 60H30; Secondary 35J60, 60J60
• Key words and phrases. Positive solutions of semilinear elliptic PDEs, superdiffusions, conditional diffusions, $\mathbb{N}$-measures
• Received by editors April 23, 2004
• Posted on August 2, 2004
• Communicated by Mark Freidlin
• Comments (When Available)

E. B. Dynkin

Department of Mathematics, Cornell University, Ithaca, NY 14853

E-mail address: ebd1@cornell.edu

Partially supported by the National Science Foundation Grant DMS-0204237