Moderate solutions of semilinear elliptic equations with Hardy potential

Abstract

Let $Ω$ be a bounded smooth domain in $\mathbb{R}^N$. We study positive solutions of equation (E) $-L_μu+ u^q = 0$ in $Ω$ where $L_μ=Δ+ \fracμ{δ^2}$, $0<μ$, $q>1$ and $δ(x)=\mathrm{dist}\,(x,\partialΩ)$. A positive solution of (E) is moderate if it is dominated by an $L_μ$-harmonic function. If $μ<C_H(Ω)$ (the Hardy constant for $Ω$) every positive $L_μ$- harmonic functions can be represented in terms of a finite measure on $\partialΩ$ 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, $1<q<q_{μ,c}$. (The critical value depends only on $N$ and $μ$.) For $q\geq q_{μ,c}$ 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 $L_μ$. These results form the basis for the study of the nonlinear problem.