<p>In this study, a class of Schrödinger–Poisson system with <i>N</i>-Laplacian operator and critical exponential growth is taken into consideration <Equation ID="Equ56"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2826_Article_Equ56.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="402" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _N u + V(x)|u|^{N-2}u + \lambda \phi |u|^{p-2}u = \mu g(u), &amp; x \in \mathbb {R}^N,\\ -\Delta \phi = |u|^p, &amp; x \in \mathbb {R}^N, \end{array}\right. } \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>N</mi> </msub> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>+</mo> <mi>λ</mi> <mi>ϕ</mi> <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> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <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 /> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </msup> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2826_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="175" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Delta _N u = \operatorname {div}(|\nabla u|^{N-2}\nabla u)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>N</mi> </msub> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mo>div</mo> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the <i>N</i>-Laplacian operator with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2826_Article_IEq2.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="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2826_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2826_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> are positive parameters, and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2826_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(p &gt; N\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>&gt;</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>. Here, <i>V</i>(<i>x</i>) and <i>g</i>(<i>u</i>) are smooth functions. With a primary technique of constrained minimization, we determine the existence, energy estimate, and convergence property of nodal (that is, sign-changing) solutions.</p>

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Nodal Solutions to Schrödinger–Poisson System with N-Laplacian Operator and Critical Exponential Growth

  • Hongling Pu,
  • Sihua Liang,
  • Shuguan Ji

摘要

In this study, a class of Schrödinger–Poisson system with N-Laplacian operator and critical exponential growth is taken into consideration \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _N u + V(x)|u|^{N-2}u + \lambda \phi |u|^{p-2}u = \mu g(u), & x \in \mathbb {R}^N,\\ -\Delta \phi = |u|^p, & x \in \mathbb {R}^N, \end{array}\right. } \end{aligned}\) - Δ N u + V ( x ) | u | N - 2 u + λ ϕ | u | p - 2 u = μ g ( u ) , x R N , - Δ ϕ = | u | p , x R N , where \(\Delta _N u = \operatorname {div}(|\nabla u|^{N-2}\nabla u)\) Δ N u = div ( | u | N - 2 u ) is the N-Laplacian operator with \(N \ge 3\) N 3 , \(\lambda \) λ and \(\mu \) μ are positive parameters, and \(p > N\) p > N . Here, V(x) and g(u) are smooth functions. With a primary technique of constrained minimization, we determine the existence, energy estimate, and convergence property of nodal (that is, sign-changing) solutions.