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

Abstract

Let $\Omega $ be a smooth bounded domain. We are concerned about the following nonlinear elliptic problem: $\Delta u + |x|^{\alpha}u^{p} = 0, u > 0 \quad$ in $\Omega$, $ u = 0 \quad$ on $\partial \Omega $, where $\alpha > 0, p \in (1,\frac{n+2}{n-2}).$ In this paper, we show that for $n \ge 8$, a maximum point $x_{\alpha }$ of a least energy solution of above problem converges to a point $x_0 \in \partial^$*$ \Omega $ satisfying $H(x_0) = \min_$$\w \in \partial ^$*$ \Omega$$H( w )$ as $\alpha \to \infty,$ where $H$ is the mean curvature on $\partial \Omega $ and $\partial ^$*$\Omega \equiv \{ x \in \partial \Omega : |x| \ge |y|$ for any $y \in \Omega \}.$