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<p>We study the uniqueness of reaction-diffusion steady states in general domains with Dirichlet boundary data. Here we consider “positive” (monostable) reactions. We describe geometric conditions on the domain that ensure uniqueness and we provide complementary examples of nonuniqueness. Along the way, we formulate a number of open problems and conjectures. To derive our results, we develop a general framework, the <i>stable-compact method</i>, to study qualitative properties of nonlinear elliptic equations.</p>

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A Stable-Compact Method for Qualitative Properties of Semilinear Elliptic Equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月26日

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  • Henri Berestycki,
  • Cole Graham

摘要

We study the uniqueness of reaction-diffusion steady states in general domains with Dirichlet boundary data. Here we consider “positive” (monostable) reactions. We describe geometric conditions on the domain that ensure uniqueness and we provide complementary examples of nonuniqueness. Along the way, we formulate a number of open problems and conjectures. To derive our results, we develop a general framework, the stable-compact method, to study qualitative properties of nonlinear elliptic equations.