<p>We investigate normalized solutions of the following Choquard equation perturbed by saturable nonlinearity <Equation ID="Equ72"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2925_Article_Equ72.gif" Format="GIF" Height="54" Rendition="HTML" Resolution="72" Type="Linedraw" Width="407" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u=\left( I_{\alpha }*|u|^{p}\right) |u|^{p-2}u+\mu \frac{g(x)+u^{2}}{1+g(x)+u^{2}}u\ \ &amp; \text { in}\ \mathbb {R}^{N}, \\ \int _{\mathbb {R}^{N}}u^{2}dx=c&gt;0, &amp; \end{array} \right. \end{aligned}\)</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> <mfenced close=")" open="("> <msub> <mi>I</mi> <mi>α</mi> </msub> <mrow /> <mo>∗</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </msup> </mfenced> <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> <mi>μ</mi> <mfrac> <mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>u</mi> <mspace width="4pt" /> <mspace width="4pt" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="4pt" /> <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> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>c</mi> <mo>&gt;</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2925_Article_IEq1.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="200" /> </InlineMediaObject> <EquationSource Format="TEX">\(2_{\alpha }:=\frac{N+\alpha }{N}\le p\le 2_{\alpha }^{*}:=\frac{N+\alpha }{N-2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mn>2</mn> <mi>α</mi> </msub> <mo>:</mo> <mo>=</mo> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>α</mi> </mrow> <mi>N</mi> </mfrac> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mmultiscripts> <mn>2</mn> <mrow> <mi>α</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>:</mo> <mo>=</mo> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>α</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2925_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu \in \mathbb {R}\backslash \{0\},\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="true">\</mo> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and <i>g</i>(<i>x</i>) is a bounded intensity function on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2925_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^{N}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>. Under different assumptions on <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2925_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(p,\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>,</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> and <i>g</i>(<i>x</i>), we prove several existence and nonexistence results. We also describe some properties on the associated Lagrange multipliers <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2925_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda ,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> including the asymptotic behavior as <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2925_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(c\rightarrow 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and the relationship with the distribution potential <i>g</i>(<i>x</i>).</p>

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Choquard equations with saturable reaction

  • Juntao Sun,
  • Jian Zhang,
  • Vicenţiu D. Rǎdulescu,
  • Tsung-fang Wu

摘要

We investigate normalized solutions of the following Choquard equation perturbed by saturable nonlinearity \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u=\left( I_{\alpha }*|u|^{p}\right) |u|^{p-2}u+\mu \frac{g(x)+u^{2}}{1+g(x)+u^{2}}u\ \ & \text { in}\ \mathbb {R}^{N}, \\ \int _{\mathbb {R}^{N}}u^{2}dx=c>0, & \end{array} \right. \end{aligned}\) - Δ u + λ u = I α | u | p | u | p - 2 u + μ g ( x ) + u 2 1 + g ( x ) + u 2 u in R N , R N u 2 d x = c > 0 , where \(2_{\alpha }:=\frac{N+\alpha }{N}\le p\le 2_{\alpha }^{*}:=\frac{N+\alpha }{N-2}\) 2 α : = N + α N p 2 α : = N + α N - 2 , \(\mu \in \mathbb {R}\backslash \{0\},\) μ R \ { 0 } , and g(x) is a bounded intensity function on \(\mathbb {R}^{N}\) R N . Under different assumptions on \(p,\mu \) p , μ and g(x), we prove several existence and nonexistence results. We also describe some properties on the associated Lagrange multipliers \(\lambda ,\) λ , including the asymptotic behavior as \(c\rightarrow 0\) c 0 and the relationship with the distribution potential g(x).