A New Inequality for Superdiffusions and Its Applications to Nonlinear Differential Equations
نویسنده
چکیده
Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation Lu = uα with α > 1 in a bounded smooth domain E ⊂ Rd. In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called σ-moderate. The question if all solutions belong to this class remained open. In 2002 Mselati proved that this is true for the equation ∆u = u2 in a domain of class C4. His principal tool— the Brownian snake—is not applicable to the case α 6= 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.
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