On upper bounds for positive solutions of semilinear equations

Suppose that E is a bounded domain of class C^2,λ in and L is a uniformly elliptic operator in E. The set of all positive solutions of the equation Lu=ψ(u) in E was investigated by a number of authors for various classes of functions ψ. In Dynkin and Kuznetsov (Comm. Pure Appl. Math. 51 (1998) 897) we defined, for every Borel subset Γ of ∂E, two such solutions u_Γ?w_Γ. We also introduced a class of solutions u_ν in 1-1 correspondence with a certain class of σ-finite measures ν on ∂E. With every we associated a pair (Γ,ν) where Γ is a Borel subset of ∂E and . We called this pair the fine boundary trace of u and we denoted in tr(u).Let u⊕v stand for the maximal solution dominated by u+v. We say that u belongs to the class if the condition tr(u)=(Γ,ν) implies that u?w_Γ⊕u_ν and we say that u belongs to if the condition tr(u)=(Γ,ν) implies that u?u_Γ⊕u_ν.It was proved in Dynkin and Kuznetsov (1998) that, under minimal assumptions on L and ψ, the class contains all bounded domains. It follows from results of Mselati (Thése de Doctorat de l'Université Paris 6, 2002; C.R. Acad. Sci. Paris Sér. I 332 (2002); Mem. Amer. Math. Soc. (2003), to appear), that all E of the class C⁴ belong to where Δ is the Laplacian and ψ(u)=u². [Mselati proved that, in his case, u_Γ=w_Γ and therefore the condition tr(u)=(Γ,ν) implies u=u_Γ⊕u_ν=w_Γ⊕u_ν.]By modifying Mselati's arguments, we extend his result to ψ(u)=u^α with 1<α?2 and all bounded domains of class C^2,λ.We start from proving a general localization theorem: under broad assumptions on L, ψ if, for every y∈∂E there exists a domain such that E′⊂E and ∂E∩∂E′ contains a neighborhood of y in ∂E.