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<p>We study the Hénon equation in a reflectionally symmetric or point symmetric domain <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2970_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, which is unbounded and it is getting narrower near the infinity. We call <i>u</i>(<i>x</i>) a least energy solution if it is a minimizer of the Rayleigh quotient corresponding to the Hénon equation. We shall show that no least energy solution is reflectionally symmetric and even. Furthermore, we prove the existence of a positive solution which has the exact symmetry of reflection.</p>

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Reflectionally asymmetric solutions of the Hénon equation in unbounded domains

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  • 发表日期：

    2025年5月12日

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  • Ryuji Kajikiya

摘要

We study the Hénon equation in a reflectionally symmetric or point symmetric domain \(\Omega \) Ω , which is unbounded and it is getting narrower near the infinity. We call u(x) a least energy solution if it is a minimizer of the Rayleigh quotient corresponding to the Hénon equation. We shall show that no least energy solution is reflectionally symmetric and even. Furthermore, we prove the existence of a positive solution which has the exact symmetry of reflection.