<p>In this paper, we study the Kirchhoff equation with Choquard nonlinearity of the form <Equation ID="Equ21"> <EquationSource Format="TEX">$\begin{aligned} -\left( a+b \int _{\mathbb {Z}^3}|\nabla u|^{2} d \mu \right) \Delta u+h(x) u=\left( R_{\alpha }*|u|^{p}\right) |u|^{p-2}u,\quad x\in \mathbb {Z}^3, \end{aligned}$</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> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <mfenced close=")" open="("> <msub> <mi>R</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> <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"> <EquationSource Format="TEX">$a,\,b>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>b</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha \in (0,3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are constants and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> represents the Green’s function of the discrete fractional Laplacian that behaves as the Riesz potential. Under different assumptions on potential function <i>h</i>, we first establish the existence of ground state solutions for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p>2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> based on the Nehari manifold. Subsequently, we obtain the existence of ground state sign-changing solutions for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p>4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> by adopting constrained minimization arguments on the sign-changing Nehari manifold.</p>

Solutions to discrete nonlinear Kirchhoff–Choquard equations with power nonlinearity

期刊论文
发表日期：

2025年7月13日

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Lidan Wang

摘要

In this paper, we study the Kirchhoff equation with Choquard nonlinearity of the form $\begin{aligned} -\left( a+b \int _{\mathbb {Z}^3}|\nabla u|^{2} d \mu \right) \Delta u+h(x) u=\left( R_{\alpha }*|u|^{p}\right) |u|^{p-2}u,\quad x\in \mathbb {Z}^3, \end{aligned}$ $\begin{matrix} - (a + b \int_{Z^{3}} {| \nabla u |}^{2} d μ) Δ u + h (x) u = (R_{α} * {| u |}^{p}) {| u |}^{p - 2} u, x \in Z^{3}, \end{matrix}$ where $a,\,b>0$ $a, b > 0$ , $\alpha \in (0,3)$ $α \in (0, 3)$ are constants and $R_{\alpha }$ $R_{α}$ represents the Green’s function of the discrete fractional Laplacian that behaves as the Riesz potential. Under different assumptions on potential function h, we first establish the existence of ground state solutions for $p>2$ $p > 2$ based on the Nehari manifold. Subsequently, we obtain the existence of ground state sign-changing solutions for $p>4$ $p > 4$ by adopting constrained minimization arguments on the sign-changing Nehari manifold.