Moderate solutions of semilinear elliptic equations with Hardy potential

Let Omega be a bounded smooth domain in R-N. We study positive solutions of equation (E) - L(mu)u + u(q) = 0 in Omega where L-mu = Delta + mu/delta(2), 0 < mu, q > 1 and delta(x) = dist (x, partial derivative Omega). A positive solution of (E) is moderate if it is dominated by an L-mu-harmonic function. If mu < C-H (Omega) (the Hardy constant for Omega) every positive L-mu-harmonic function can be represented in terms of a finite measure on partial derivative Omega via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary trace by which we obtain a complete classification of the positive moderate solutions of (E) in the subcritical case, 1 < q < q(mu,c). (The critical value depends only on N and mu) For q >= q(mu,c) there exists no moderate solution with an isolated singularity on the boundary. The normalized boundary trace and associated boundary value problems are also discussed in detail for the linear operator L-mu. These results form the basis for the study of the nonlinear problem. (C) 2015 Elsevier Masson SAS. All rights reserved.