The boundary trace of positive solutions of semilinear elliptic equations: The supercritical case

We study the boundary trace problem for solutions of the equation (E) -Delta u + \u\(q-1 )u = 0 in a smooth bounded domain Omega in the supercritical case q greater than or equal to (N + 1)/(N - 1). A bounded Borel measure nu on partial derivative Omega, not necessarily positive, is a q-trace if there exists a solution of (E) with boundary trace nu. It is known that the solution is unique. In the first part of the paper we provide a characterization of the space of q-traces in terms of Bessel potentials. In the second part we consider arbitrary positive solutions of (E). Each such solution has a well defined boundary trace which can be represented by a positive, not necessarily bounded, outer regular Borel measure (see [22, 24]). We provide necessary and sufficient conditions on such a measure nu in order that there exists a solution of (E) with trace nu. It is shown that in this case the solution of the boundary value problem may not be unique, (see also [19]). (C) Elsevier, Paris.