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<p>We study normalised solutions for a Choquard equation in the plane with polynomial Riesz kernel and exponential nonlinearities, which are critical in the sense of Trudinger–Moser. For all prescribed values of the mass, we prove existence of a positive radial solution by a variational argument, which exploits a delicate analysis on the mountain pass level. Under an additional monotonicity assumption on the nonlinearity, such a solution turns out to be also a ground state in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2478_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^1(\mathbb {R}^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Our work extends the results by Dou et al. (J Geom Anal 34(10):317, 2024) to the Choquard setting, improving in several directions those by Deng and Yu (Z Angew Math Phys 74(3):103, 2023).</p>

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Positive solutions with prescribed mass for a planar Choquard equation with critical growth

  • 期刊论文

  • 发表日期：

    2025年4月18日

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  • Ling Huang,
  • Giulio Romani

摘要

We study normalised solutions for a Choquard equation in the plane with polynomial Riesz kernel and exponential nonlinearities, which are critical in the sense of Trudinger–Moser. For all prescribed values of the mass, we prove existence of a positive radial solution by a variational argument, which exploits a delicate analysis on the mountain pass level. Under an additional monotonicity assumption on the nonlinearity, such a solution turns out to be also a ground state in \(H^1(\mathbb {R}^2)\) H 1 ( R 2 ) . Our work extends the results by Dou et al. (J Geom Anal 34(10):317, 2024) to the Choquard setting, improving in several directions those by Deng and Yu (Z Angew Math Phys 74(3):103, 2023).