A class of boundary traces for solutions of the equation Lu=ψ(u)

Our subject is the class U of all positive solutions of a semilinear equation Lu=ψ(u) in E where L is a second order elliptic differential operator, E is a domain in Rd and ψ belongs to a convex class Ψ of C1 functions which contains functions ψ(u)=uα with α>1. A special role is played by a class U0 of solutions which we call σ-moderate. A solution u is moderate if there exists h⩾u such that Lh=0 in E. We say that u∈U is σ-moderate if u is the limit of an increasing sequence of moderate solutions. In [E.B. Dynkin, S.E. Kuznetsov, Fine topology and fine trace on the boundary associated with a class of quasilinear differential equations, Comm. Pure Appl. Math. 51 (1998) 897–936] all σ-moderate solutions were classified by using their fine boundary traces.2 In [M. Marcus, L. Véron, The precise boundary trace of positive solutions of the equation Δu=uq in the supercritical case, in: Perspectives in Nonlinear Partial Differential Equations, in honor of Haim Brezis, Contemp. Math., Amer. Math. Soc., Providence, RI, 2007, arxiv.org/math/0610102] Marcus and Véron introduced a different characteristic (called the precise trace) for solutions of the equation Δu=uα with α>1 in a bounded C2 domain. In the present paper we develop a general scheme covering both approaches and we prove the equivalence, in a certain sense, of the fine and precise traces. An implication of this equivalence is a Wiener type criterion for vanishing of the Poisson kernel of the equation Lu(x)=a(x)u(x).