On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent

Futoshi Takahashi

doi:10.3934/cpaa.2013.12.1237

Article Contents

2013, Volume 12, Issue 3: 1237-1241. Doi: 10.3934/cpaa.2013.12.1237

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On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent

Futoshi Takahashi^1,

1.
Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585

Received: December 2011

Revised: May 2012

Published: May 2013

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Abstract

In this note, we prove that least energy solutions of the two-dimensional Hénon equation \begin{eqnarray*} -\Delta u = |x|^{2\alpha} u^p \quad x \in \Omega, \quad u 0 \quad x \in \Omega, \quad u = 0 \quad x \in \partial \Omega, \end{eqnarray*} where $\Omega$ is a smooth bounded domain in $R^2$ with $0 \in \Omega$, $\alpha \ge 0$ is a constant and $p >1$, have only one global maximum point when $\alpha > e-1$ and the nonlinear exponent $p$ is sufficiently large. This answers positively to a recent conjecture by C. Zhao (preprint, 2011).

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Mathematics Subject Classification: Primary: 35B40, 35J20, 35J25.

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References

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