<p>In this paper, we focus on the solutions to the following Schrödinger equations with van der Waals type potentials <Equation ID="Equ102"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2572_Article_Equ102.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="475" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} -\Delta u+V(x)u=\lambda u+\mu (|x|^{-\alpha }*|u|^{2})u+(|x|^{-4}*|u|^{2})u,~~~~ x\in {\mathbb {R}}^{N} \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <mi>λ</mi> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>α</mi> </mrow> </msup> <msup> <mrow> <mrow /> <mo>∗</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mo stretchy="false">)</mo> <mi>u</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </msup> <msup> <mrow> <mrow /> <mo>∗</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">)</mo> <mi>u</mi> <mo>,</mo> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi>x</mi> <mo>∈</mo> </mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with prescribed mass <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2572_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">\(\int _{{\mathbb {R}}^{N}}|u|^{2}dx=c^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <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> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2572_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(N\geqslant 5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>⩾</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2572_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu , c&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>,</mo> <mi>c</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2572_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(0&lt;\alpha &lt;4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>α</mi> <mo>&lt;</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, <i>V</i> is an external potential vanishing at infinity, and the parameter <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2572_Article_IEq5.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. Under some explicit assumptions on <i>V</i>, we prove the existence of normalized solutions for the above problem.</p>

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Normalized solutions to the Schrödinger equations with van der Waals type potentials

  • Zhi-Jie Wang,
  • Hong-Rui Sun

摘要

In this paper, we focus on the solutions to the following Schrödinger equations with van der Waals type potentials \(\begin{aligned} -\Delta u+V(x)u=\lambda u+\mu (|x|^{-\alpha }*|u|^{2})u+(|x|^{-4}*|u|^{2})u,~~~~ x\in {\mathbb {R}}^{N} \end{aligned}\) - Δ u + V ( x ) u = λ u + μ ( | x | - α | u | 2 ) u + ( | x | - 4 | u | 2 ) u , x R N with prescribed mass \(\int _{{\mathbb {R}}^{N}}|u|^{2}dx=c^{2}\) R N | u | 2 d x = c 2 , where \(N\geqslant 5\) N 5 , \(\mu , c>0\) μ , c > 0 , \(0<\alpha <4\) 0 < α < 4 , V is an external potential vanishing at infinity, and the parameter \(\lambda \in {\mathbb {R}}\) λ R appears as a Lagrange multiplier. Under some explicit assumptions on V, we prove the existence of normalized solutions for the above problem.