In this paper, we study the discrete logarithmic Kirchhoff equation <Equation ID="Equ29"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2101_Article_Equ29.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="496" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} -\left( a+b \int _{\mathbb {Z}^3}|\nabla u|^{2} d \mu \right) \Delta u+(\lambda h(x)+1) u=|u|^{p-2}u \log u^{2}, \quad x\in \mathbb {Z}^3, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <mfenced close=")" open="("> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>3</mn> </msup> </msub> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>μ</mi> </mfenced> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <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>log</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>3</mn> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2101_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$a,b>0, p>6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mi>p</mi> <mo>></mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2101_Article_IEq2.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> is a positive parameter. Under suitable assumptions on h(x), we prove the existence and asymptotic behavior of least energy sign-changing solutions for the equation by the method of Nehari manifold.

Sign-changing solutions to discrete nonlinear logarithmic Kirchhoff equations

期刊论文
发表日期：

2025年7月15日

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Lidan Wang

摘要

In this paper, we study the discrete logarithmic Kirchhoff equation $\begin{aligned} -\left( a+b \int _{\mathbb {Z}^3}|\nabla u|^{2} d \mu \right) \Delta u+(\lambda h(x)+1) u=|u|^{p-2}u \log u^{2}, \quad x\in \mathbb {Z}^3, \end{aligned}$ $\begin{matrix} - (a + b \int_{Z^{3}} {| \nabla u |}^{2} d μ) Δ u + (λ h (x) + 1) u = {| u |}^{p - 2} u log u^{2}, x \in Z^{3}, \end{matrix}$ where $a,b>0, p>6$ $a, b > 0, p > 6$ and $\lambda $ $λ$ is a positive parameter. Under suitable assumptions on h(x), we prove the existence and asymptotic behavior of least energy sign-changing solutions for the equation by the method of Nehari manifold.