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

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.