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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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