TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 11, November 1998, Pages 4521–4552 S 0002-9947(98)02006-6 SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS ON A RIEMANNIAN MANIFOLD AND THEIR TRACE ON THE MARTIN BOUNDARY E. B. DYNKIN AND S. E. KUZNETSOV Abstract. Let L be a second order elliptic differential operator on a Rie- mannian manifold E with no zero order terms. We say that a function h is L-harmonic if Lh = 0. Every positive L-harmonic function has a unique representation h(x)=  E ′ k(x, y)ν(dy), where k is the Martin kernel, E ′ is the Martin boundary and ν is a finite measure on E ′ concentrated on the minimal part E ∗ of E ′ . We call ν the trace of h on E ′ . Our objective is to investigate positive solutions of a nonlinear equation Lu = u α on E (*) for 1 <α ≤ 2 [the restriction α ≤ 2 is imposed because our main tool is the (L, α)-superdiffusion, which is not defined for α> 2]. We associate with every solution u of (*) a pair (Γ,ν), where Γ is a closed subset of E ′ and ν is a Radon measure on O = E ′ \ Γ. We call (Γ,ν) the trace of u on E ′ . Γ is empty if and only if u is dominated by an L-harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. In an earlier paper, we investigated the case when L is a second order elliptic differential operator in R d and E is a bounded smooth domain in R d . We obtained necessary and sufficient conditions for a pair (Γ,ν) to be a trace, and we gave a probabilistic formula for the maximal solution with a given trace. The general theory developed in the present paper is applicable, in partic- ular, to elliptic operators L with bounded coefficients in an arbitrary bounded domain of R d , assuming only that the Martin boundary and the geometric boundary coincide. 0. Introduction 0.1 Definition of the trace. The key ingredients of this definition are: (1) a concept of a moderate solution and (2) operators Q B indexed by closed subsets of the Martin boundary E ′ and acting on positive solutions of the equation Lu = u α . (0.1) Received by the editors August 6, 1996. 1991 Mathematics Subject Classification. Primary 60J60, 35J60; Secondary 60J80, 60J45, 35J65. Partially supported by National Science Foundation Grant DMS-9301315. c 1998 American Mathematical Society 4521 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use