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

Abstract

Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation L u = u α L u=u^\alpha with α > 1 \alpha >1 in a bounded smooth domain E ⊂ R d E\subset \mathbb {R}^d . In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called σ \sigma -moderate. The question if all solutions belong to this class remained open. In 2002 Mselati proved that this is true for the equation Δ u = u 2 \Delta u=u^2 in a domain of class C 4 C^4 . His principal tool—the Brownian snake—is not applicable to the case α ≠ 2 \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.