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<p>We consider the classical geometric problem of prescribing the scalar and the boundary mean curvature problem in the unit ball <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1102_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {B}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> endowed with the standard Euclidean metric. We will deal with the case of negative scalar curvature showing the existence of infinitely many non-radial positive solutions when <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1102_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(N\ge 5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. This is the first result of existence of solutions in the case of negative prescribed scalar curvature problem in higher dimensions.</p>

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Infinitely many solutions for a boundary Yamabe problem

  • 期刊论文

  • 发表日期：

    2025年7月12日

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  • Luca Battaglia,
  • Yixing Pu,
  • Giusi Vaira

摘要

We consider the classical geometric problem of prescribing the scalar and the boundary mean curvature problem in the unit ball \({\mathbb {B}}^n\) B n endowed with the standard Euclidean metric. We will deal with the case of negative scalar curvature showing the existence of infinitely many non-radial positive solutions when \(N\ge 5\) N ≥ 5 . This is the first result of existence of solutions in the case of negative prescribed scalar curvature problem in higher dimensions.