<p>This paper investigates the existence and concentration behavior of nodal solutions for the planar Schrödinger-Poisson system with subcritical exponential growth <Equation ID="Equ34"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10289_Article_Equ34.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="424" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} \left\{ \begin{array}{l} -\Delta u+V(\varepsilon x)u+\mu \phi u= f(u)\ \ \text{ in }\ {\mathbb {R}}^2,\\ \Delta \phi =u^2\ \ \text{ in }\ {\mathbb {R}}^2, \end{array}\right. \qquad \qquad \qquad ({\mathcal {S}}) \end{aligned}\)</EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10289_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon, \mu &gt;0\)</EquationSource> </InlineEquation> are parameters, <i>V</i> and <i>f</i> are continuous functions. Under suitable assumptions, the existence of nodal solutions is established, using variational methods. Furthermore, we prove that the nodal solutions of (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10289_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathcal {S}}\)</EquationSource> </InlineEquation>) concentrate around the minimum point of <i>V</i> and exhibit exponential decay at infinity.</p>

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Concentration Phenomena of Sign-Changing Solutions for the Planar Schrödinger-Poisson Systems

  • Yiqing Li,
  • Patrizia Pucci,
  • Binlin Zhang

摘要

This paper investigates the existence and concentration behavior of nodal solutions for the planar Schrödinger-Poisson system with subcritical exponential growth \(\begin{aligned} \left\{ \begin{array}{l} -\Delta u+V(\varepsilon x)u+\mu \phi u= f(u)\ \ \text{ in }\ {\mathbb {R}}^2,\\ \Delta \phi =u^2\ \ \text{ in }\ {\mathbb {R}}^2, \end{array}\right. \qquad \qquad \qquad ({\mathcal {S}}) \end{aligned}\) where \(\varepsilon, \mu >0\) are parameters, V and f are continuous functions. Under suitable assumptions, the existence of nodal solutions is established, using variational methods. Furthermore, we prove that the nodal solutions of ( \({\mathcal {S}}\) ) concentrate around the minimum point of V and exhibit exponential decay at infinity.