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 $L_\mu=\Delta + \frac{\mu}{\delta^2}$, $0<\mu$, $q>1$ and $\delta(x)=\mathrm{dist}\,(x,\partial\Omega)$. 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 functions 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 $q\geq q_{\mu,c}$ 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.