Moderate solutions of semilinear elliptic equations with Hardy potential

Abstract

Let Ω be a bounded smooth domain in RN. We study positive solutions of equation (E) −Lμu+uq=0 in Ω where Lμ=Δ+μδ2, 0<μ, q>1 and δ(x)=dist(x,∂Ω). A positive solution of (E) is moderate if it is dominated by an Lμ-harmonic function. If μ<CH(Ω) (the Hardy constant for Ω) every positive Lμ-harmonic function can be represented in terms of a finite measure on ∂Ω 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<qμ,c. (The critical value depends only on N and μ.) For q≥qμ,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 Lμ. These results form the basis for the study of the nonlinear problem.