Moderate solutions of semilinear elliptic equations with Hardy potential

Abstract

Let $\Omega$ be a bounded smooth domain in ${\mathbb{R}}^N$. We study positive solutions of equation (E) $-L_{\mu }u+u^q=0$ in $\Omega$ where $L_{\mu }=\mathrm{\Delta }+\frac{\mu }{{\delta }^2}$, $0<\mu$, $q>1$ and $\delta (x)=\mathrm{dist}(x,\partial \mathrm{\Omega })$. A positive solution of (E) is moderate if it is dominated by an $L_{\mu }$-harmonic function. If $\mu <C_H(\mathrm{\Omega })$ (the Hardy constant for $\Omega$) every positive $L_{\mu }$-harmonic function can be represented in terms of a finite measure on $\partial \Omega$ 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 $q\ge q_{\mu ,c}$ 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.