FINE TOPOLOGY AND FINE TRACE ON THE BOUNDARY ASSOCIATED WITH A CLASS OF SEMILINEAR DIFFERENTIAL-EQUATIONS

We investigate the set of all positive solutions of a semilinear equat ion Lu = psi(u) where L is a second-order elliptic differential operat or in a domain E of R-d or, more generally, in a Riemannian manifold a nd psi belongs to a wide class of convex functions that contains psi(u ) = u(alpha) for all alpha > 1. We define boundary singularities of a solution u, in terms of points of rapid growth of the right derivative psi(+)(u), we introduce a fine topology and a fine trace of u on the Martin boundary, and we construct the minimal solution for every possi ble value of this trace. (C) 1998 John Wiley & Sons, Inc.