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<p>We consider a class of non–linear and non–local functionals giving rise to the Choquard equation with a suitably regular interaction potential, modelling, i.a., gases with impurities and axion stars. We study how existence of minimizers depends on the coupling constant, and find that there is a critical interaction strength needed for the minimizers to exist, both in dimensions two and three. In <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(d=3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, a minimizer exists also at the critical coupling but none do in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> under suitable assumptions on the potential. We also establish that in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(d=3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> there exist other critical points beyond the global minimizer.</p>

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Existence and order of the self–binding transition in non–local non–linear Schrödinger equations

  • 期刊论文

  • 发表日期：

    2026年3月7日

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  • Norihisa Ikoma,
  • Krzysztof Myśliwy

摘要

We consider a class of non–linear and non–local functionals giving rise to the Choquard equation with a suitably regular interaction potential, modelling, i.a., gases with impurities and axion stars. We study how existence of minimizers depends on the coupling constant, and find that there is a critical interaction strength needed for the minimizers to exist, both in dimensions two and three. In \(d=3\) d = 3 , a minimizer exists also at the critical coupling but none do in \(d=2\) d = 2 under suitable assumptions on the potential. We also establish that in \(d=3\) d = 3 there exist other critical points beyond the global minimizer.