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A new inequality for superdiffusions and its applications to nonlinear differential equations
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## A new inequality for superdiffusions and its applications to nonlinear differential equations

### E. B. Dynkin

**Abstract.**
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

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