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

Jaeyoung Byeon; Sungwon Cho; Junsang Park

doi:10.3934/dcds.2011.30.1055

Article Contents

2011, Volume 30, Issue 4: 1055-1081. Doi: 10.3934/dcds.2011.30.1055

This issue Previous Article Asymptotics for a generalized Cahn-Hilliard equation with forcing terms Next Article Radial symmetry of solutions for some integral systems of Wolff type

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

1.
Department of Mathematics & PMI, POSTECH, Pohang, Kyungbuk 790-784
2.
Department of Mathematics Education, Gwangju National University of Education, 93 Pilmunlo Bugku, Gwangju 500-703
3.
Department of Mathematics, POSTECH, Pohang, Kyungbuk 790-784

Received: April 30, 2010

Revised: January 31, 2011

Published: September 2011

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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 $y \in \Omega \}.$

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Mathematics Subject Classification: Primary: 35J60, 35J20; Secondary: 35J25, 47J30.

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