Asymptotic behavior of solutions to semilinear elliptic equations with Hardy potential

Abstract:

Let $\Omega$ be an open bounded domain in $\mathbb {R}^N (N\geq 3)$ with smooth boundary $\partial \Omega$, $0\!\in \!\Omega$. We are concerned with the asymptotic behavior of solutions for the elliptic problem: \begin{equation*} (*)\qquad \qquad \qquad \ -\Delta u-\frac {\mu u}{|x|^2}=f(x, u),\qquad u\in H^1_0(\Omega ),\qquad \qquad \qquad \qquad \ \ \end{equation*} where $0\leq \mu <\big (\frac {N-2}{2}\big )^2$ and $f(x, u)$ satisfies suitable growth conditions. By Moser iteration, we characterize the asymptotic behavior of nontrivial solutions for problem $(*)$. In particular, we point out that the proof of Proposition 2.1 in Proc. Amer. Math. Soc. 132 (2004), 3225–3229, is wrong.