On upper bounds for positive solutions of semilinear equations

摘要

Suppose that E is a bounded domain of class C-2lambda in R-d and L is a uniformly elliptic operator ill E. The set U of all positive solutions of the equation Lu = psi(u) ill E was investigated by a number of authors for various classes of functions psi. In Dynkin and Kuznetsov (Comm. Pure Appl. Math. 51 (1998) 897) we defined, for every Borel subset Gamma of partial derivativeE, two such solutions u(Gamma) < w(Gamma). We also introduced a class of solutions u(v) in 1-1 correspondence with a certain class N-0 of sigma-finite measures v oil partial derivativeE. With every u epsilon U we associated a pair (Gamma, v) where Gamma is a Borel subset of partial derivativeE and v epsilon N-0. We called this pair the fine boundary trace of it and we denoted in tr(u).Let it u circle plus a stand for the maximal solution dominated by it + u. We say that it belongs to the class E-L.psi if the condition tr(u) (Gamma, v) implies that u less than or equal to w(Gamma) circle plus u(v) and we say that it belongs to E-L.psi if the condition tr(u) (Gamma, v) implies that u less than or equal to w(Gamma) circle plus u(v).It was proved ill Dynkin and Kuznetsov (1998) that, under minimal assumptions on L and psi, the class E-L.psi(*) contains all bounded domains. It follows from results of Mselati (These de Doctorat de l'Universite Paris 6, 2002; C.R. Acad. Sci. Paris Ser. I 332 (2002); Mem. Amer. Math. Soc. (2003), to appear), that all E of the class C-4 belong E-Deltapsi to where Delta is the Laplacian and psi(u) = u(2). [Mselati proved that, in his case, u(Gamma) = w(Gamma) and therefore the ;condition tr(u) = (h, v) implies u = u(Gamma)circle plusu(v) = w(Gamma)circle plusu(v)]By modifying Mselati's arguments, we extend his result to psi(u) = u(alpha) with 1 < alpha less than or equal to 2 and all bounded domains of class C-2lambda.We start from proving a general localization theorem: E epsilon E-L.psi under broad assumptions on L, psi if, for every gamma epsilon partial derivativeE there exists a domain E' epsilon E-L.psi such that E' subset of E and partial derivativeE boolean AND partial derivativeE' contains a neighborhood of y in partial derivativeE. (C) 2003 Elsevier Inc. All rights reserved.