On the location of a peak point of a least energy solution for Hénon equation (Q2275854)

The authors consider the following nonlinear elliptic equation: \[ \Delta u+\left| x\right| ^{\alpha }u^{p}=0 \text{ in }\Omega,\text{ }u>0\text{ in }\Omega \text{, and }u=0\text{ on } \partial \Omega, \tag{P} \] where \(\alpha >0,\) \(p\in (1,\frac{n+2}{n-2})\) and \(\Omega \) is a smooth bounded domain in \(\mathbb{R}^{n}\) with \(n\geq 3\). They solve partially the conjecture of \textit{J. Byeon} and \textit{Z.\ Q. Wang}, given in [J. Differ. Equations 216, No. 1, 78--108 (2005; Zbl 1114.35070)]. More explicitly they prove that, for \(n\geq 8\), the maximum point of the least energy solution of (P) converges, up to a subsequence, to a minimum point of mean curvature on \(\partial ^{\ast }\Omega \) as \(\alpha \) goes to \(\infty \). Here \(\partial ^{\ast }\Omega =\left\{ x\in \partial\Omega : \left| x\right| \geq \left| y\right| \text{ for any }y\in \Omega \right\}\).