<p>Consider the equation <Equation ID="Equ30"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1742_Article_Equ30.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="379" /> </MediaObject> <EquationSource Format="TEX">$ -\Delta u +\lambda u=(I_{\alpha } |u|^{2_{\alpha }^})|u|^{2_{\alpha }^* -2}u +|u|^{p-2}u, \quad \text { in } \mathbb {R}^N, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>λ</mi> <mi>u</mi> <mo>=</mo> <mo stretchy="false">(</mo> </mrow> <msub> <mi>I</mi> <mi>α</mi> </msub> <msup> <mrow> <mrow /> <mo>∗</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <msubsup> <mn>2</mn> <mrow> <mi>α</mi> </mrow> <mo>∗</mo> </msubsup> </msup> <msup> <mrow> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <msubsup> <mn>2</mn> <mrow> <mi>α</mi> </mrow> <mo>∗</mo> </msubsup> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>under the normalized constraint <Equation ID="Equ31"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1742_Article_Equ31.gif" Format="GIF" Height="41" Rendition="HTML" Resolution="72" Type="Linedraw" Width="147" /> </MediaObject> <EquationSource Format="TEX">$\int _{\mathbb {R}^N} |u|^2 dx =c^2 >0,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </msub> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msup> <mi>c</mi> <mn>2</mn> </msup> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1742_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$I_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> is the Riesz potential, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1742_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">$2+\frac{4}{N}<p<2_{\alpha }^*$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>+</mo> <mfrac> <mn>4</mn> <mi>N</mi> </mfrac> <mo><</mo> <mi>p</mi> <mo><</mo> <msubsup> <mn>2</mn> <mrow> <mi>α</mi> </mrow> <mo>∗</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1742_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$c>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1742_Article_IEq4.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>, appeared as an unknown Lagrange multiplier. Using the concentration compactness lemma and the Ekeland variational principle, we obtain the multiple radially normalized solutions for the above equation.</p>

Multiple radially normalized solutions for the Choquard equation involving critical exponents

期刊论文
发表日期：

2025年5月29日

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Wenjun Xing,
Chunyu Lei

摘要

Consider the equation $ -\Delta u +\lambda u=(I_{\alpha } *|u|^{2_{\alpha }^*})|u|^{2_{\alpha }^* -2}u +|u|^{p-2}u, \quad \text { in } \mathbb {R}^N, $ $- Δ u + λ u = (I_{α} {* | u |}^{2_{α}^{*}} {) | u |}^{2_{α}^{*} - 2} u + {| u |}^{p - 2} u, in R^{N},$ under the normalized constraint $\int _{\mathbb {R}^N} |u|^2 dx =c^2 >0,$ $\int_{R^{N}} {| u |}^{2} d x = c^{2} > 0,$ where $I_{\alpha }$ $I_{α}$ is the Riesz potential, $2+\frac{4}{N}<p<2_{\alpha }^*$ $2 + \frac{4}{N} < p < 2_{α}^{*}$ , $c>0$ $c > 0$ and $\lambda \in \mathbb {R}$ $λ \in R$ , appeared as an unknown Lagrange multiplier. Using the concentration compactness lemma and the Ekeland variational principle, we obtain the multiple radially normalized solutions for the above equation.