Solutions of Semilinear Differential Equations Related to Harmonic Functions

This is an attempt to establish a link between positive solutions of semilinear equations Lu=−ψ(u) and Lv=ψ(v) where L is a second order elliptic differential operator and ψ is a positive function. The equations were investigated separately by a number of authors. We try to link them via positive solutions of a linear equation Lu=0 (we call them L-harmonic functions). Let D be an arbitrary open subset of Rd and let U(D), V(D) and H(D) stand for the sets of all positive solutions in D for three equations mentioned above. We establish a 1–1 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].