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<p>In this paper, we mainly investigate the critical points of solutions to the semilinear elliptic equations with Neumann and Robin boundary conditions on two-dimensional convex domains of Riemannian surfaces. Precisely, under some certain convexity assumptions for domains <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, we show the non-degeneracy of critical points of solutions to the corresponding boundary problem in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Omega \subset {\mathbb {S}}^2,{\mathbb {R}}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>2</mn> </msup> <mo>,</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\mathbb {H}}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> by using Chen &amp; Huang’s comparison technique (Invent Math. 67:253-259, 1982), and prove the uniqueness of critical points by continuity method and topological degree argument.</p>

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On the critical points of solutions to some semilinear elliptic equations with Neumann and Robin boundary conditions on convex domains of Riemannian surfaces

  • 期刊论文

  • 发表日期：

    2025年5月18日

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  • Haiyun Deng,
  • Massimo Grossi,
  • Huaiyu Jian,
  • Xuyong Jiang

摘要

In this paper, we mainly investigate the critical points of solutions to the semilinear elliptic equations with Neumann and Robin boundary conditions on two-dimensional convex domains of Riemannian surfaces. Precisely, under some certain convexity assumptions for domains \(\Omega \) Ω , we show the non-degeneracy of critical points of solutions to the corresponding boundary problem in \(\Omega \subset {\mathbb {S}}^2,{\mathbb {R}}^2\) Ω ⊂ S 2 , R 2 or \({\mathbb {H}}^2\) H 2 by using Chen & Huang’s comparison technique (Invent Math. 67:253-259, 1982), and prove the uniqueness of critical points by continuity method and topological degree argument.