Solutions of semilinear differential equations related to harmonic functions

This is an attempt to establish a link between positive solutions of semilinear equations Lie = -psi(r) and Le = psi(r) where L. is a second order elliptic differential operator and psi is ii positive function. The equations were investigated separately by a number of authors. We try to link them via positive solutions of a linear equation Ly = 0 (we call them L-harmonic functions). Let D be an arbitrary open subset of R-d and let H(D), P(D) and H(D) stand for the sets of all positive solutions in D for three equations mentioned above. We establish a I-I correspondence between certain subclasses of these classes. Similar results are obtained also for the corresponding parabolic equations. A probabilistic interpretation in terms of a superdiffusion is given in [1]. (C) 2000 Academic Press.