<p>In this article, we study the existence of normalized solutions to the following Choquard equation with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="32_2025_415_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-constraint: <Equation ID="Equ96"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="32_2025_415_Article_Equ96.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="305" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u-\lambda u=(I_\alpha *F(u))f(u),\ \ x\in \mathbb {R}^N,\\ \int _{\mathbb {R}^N}u^2\textrm{d}x=c, \end{array}\right. } \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>-</mo> <mi>λ</mi> <mi>u</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mi>α</mi> </msub> <mrow /> <mo>∗</mo> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="4pt" /> <mspace width="4pt" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </msub> <msup> <mi>u</mi> <mn>2</mn> </msup> <mtext>d</mtext> <mi>x</mi> <mo>=</mo> <mi>c</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="32_2025_415_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(N\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="32_2025_415_Article_IEq5.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(c&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is given in advance, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="32_2025_415_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">\(I_\alpha : \mathbb {R}^N\rightarrow \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>α</mi> </msub> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is the Riesz potential of order <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="32_2025_415_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \in (0,N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the unknown parameter <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="32_2025_415_Article_IEq8.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda \in \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> appears as a Lagrange multiplier. We weaken the previous <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="32_2025_415_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-supercritical conditions and develop robust arguments to establish the existence of normalized solutions to the above equation for any <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="32_2025_415_Article_IEq10.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(c &gt; 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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The Existence of Normalized Solutions of the \(L^2\)-Supercritical Choquard Equations

  • Peng Jin,
  • Xianhua Tang,
  • Die Hu

摘要

In this article, we study the existence of normalized solutions to the following Choquard equation with \(L^2\) L 2 -constraint: \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u-\lambda u=(I_\alpha *F(u))f(u),\ \ x\in \mathbb {R}^N,\\ \int _{\mathbb {R}^N}u^2\textrm{d}x=c, \end{array}\right. } \end{aligned}\) - Δ u - λ u = ( I α F ( u ) ) f ( u ) , x R N , R N u 2 d x = c , where \(N\ge 3\) N 3 , \(c>0\) c > 0 is given in advance, \(I_\alpha : \mathbb {R}^N\rightarrow \mathbb {R}\) I α : R N R is the Riesz potential of order \(\alpha \in (0,N)\) α ( 0 , N ) and the unknown parameter \(\lambda \in \mathbb {R}\) λ R appears as a Lagrange multiplier. We weaken the previous \(L^2\) L 2 -supercritical conditions and develop robust arguments to establish the existence of normalized solutions to the above equation for any \(c > 0\) c > 0 .