Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary

Let L be a second order elliptic differential operator on a Riemannian 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) = integral(E') k(x,y)nu(dy), where k is the Martin kernel, E' is the Martin boundary and nu is a finite measure on E' concentrated on the minimal part E* of E'. We call nu the trace of h on E'. Our objective is to investigate positive solutions of a nonlinear equation (*) Lu = u(alpha) on E for 1 < alpha less than or equal to 2 [the restriction alpha less than or equal to 2 is imposed because our main tool is the (L,alpha)-superdiffusion, which is not defined for alpha > 2]. We associate with every solution u of (*) a pair (Gamma, nu), where Gamma is a closed subset of E' and nu is a Radon measure on O = E' \ Gamma. We call (Gamma, nu) the trace of u on E'. Gamma 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 (Gamma, nu) 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 particular, 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.