Asymptotic behavior of solutions to semilinear elliptic equations with Hardy potential

Abstract

Let Ω \Omega be an open bounded domain in R N ( N ≥ 3 ) \mathbb {R}^N (N\geq 3) with smooth boundary ∂ Ω \partial \Omega , 0 ∈ Ω 0\!\in \!\Omega . We are concerned with the asymptotic behavior of solutions for the elliptic problem: ( ∗ ) − Δ u − μ u | x | 2 = f ( x , u ) , u ∈ H 0 1 ( Ω ) , \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 ≤ μ > ( N − 2 2 ) 2 0\leq \mu >\big (\frac {N-2}{2}\big )^2 and f ( x , u ) 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.