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<p>In this paper, we study the existence of solutions to a type of super-Liouville equation on the compact Riemann surface <i>M</i> with boundary and with its Euler characteristic <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3024_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi (M)&lt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>&lt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. The boundary condition couples a Neumann condition for functions and a chiral boundary condition for spinors. Due to the generality of the equation, we introduce a weighted Dirac operator based on the solution to a related Liouville equation. Then we construct a Nehari manifold according to the spectral decomposition of the weighted Dirac operator, and use minimax theory on this Nehari manifold to show the existence of the non-trivial solutions.</p>

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Existence of Solutions to a super-Liouville equation with Boundary Conditions

  • 期刊论文

  • 发表日期：

    2025年5月28日

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  • Mingyang Han,
  • Ruijun Wu,
  • Chunqin Zhou

摘要

In this paper, we study the existence of solutions to a type of super-Liouville equation on the compact Riemann surface M with boundary and with its Euler characteristic \(\chi (M)<0\) χ ( M ) < 0 . The boundary condition couples a Neumann condition for functions and a chiral boundary condition for spinors. Due to the generality of the equation, we introduce a weighted Dirac operator based on the solution to a related Liouville equation. Then we construct a Nehari manifold according to the spectral decomposition of the weighted Dirac operator, and use minimax theory on this Nehari manifold to show the existence of the non-trivial solutions.