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<p>We consider a constrained minimization problem arising in the fractional Schrödinger equation with a trapping potential. By exploring some delicate energy estimates and studying decay properties of solution sequences, we obtain the concentration behavior of each minimizer of the fractional Schrödinger energy functional when <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_511_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">\(a\nearrow a^{\ast}:=\|Q\|_{2}^{2s}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>a</mi> <mo stretchy="false">↗</mo> <msup> <mi>a</mi> <mrow> <mo>∗</mo> </mrow> </msup> <mo>:=</mo> <mo>∥</mo> <mi>Q</mi> <msubsup> <mo fence="false" stretchy="false">∥</mo> <mrow> <mn>2</mn> </mrow> <mrow> <mn>2</mn> <mi>s</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation>, where <i>Q</i> is the unique positive radial solution of (−Δ)<sup><i>s</i></sup><i>u</i> + <i>su</i> − ∣<i>u</i>∣<sup>2<i>s</i></sup><i>u</i> = 0 in ℝ<sup>2</sup>. Based on the discussion of the concentration phenomenon, we prove the local uniqueness of minimizers by establishing a local Pohožaev identity and studying the blow-up estimates to the nonlocal operator (−Δ)<sup><i>s</i></sup>.</p>

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Concentration and uniqueness of minimizers for fractional Schrödinger energy functionals

  • 期刊论文

  • 发表日期：

    2025年8月4日

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  • Lintao Liu,
  • Shuai Yao,
  • Kaimin Teng,
  • Haibo Chen

摘要

We consider a constrained minimization problem arising in the fractional Schrödinger equation with a trapping potential. By exploring some delicate energy estimates and studying decay properties of solution sequences, we obtain the concentration behavior of each minimizer of the fractional Schrödinger energy functional when \(a\nearrow a^{\ast}:=\|Q\|_{2}^{2s}\) a ↗ a ∗ := ∥ Q ∥ 2 2 s , where Q is the unique positive radial solution of (−Δ)su + su − ∣u∣2su = 0 in ℝ2. Based on the discussion of the concentration phenomenon, we prove the local uniqueness of minimizers by establishing a local Pohožaev identity and studying the blow-up estimates to the nonlocal operator (−Δ)s.