We study the following equation with singular nonlinearity on locally finite graphs <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$G=(V,E)$</EquationSource> </InlineEquation>: <Equation ID="Equa"> <EquationSource Format="MATHML"><math> <mrow> <mo>{</mo> <mtable columnalign="right left right left" columnspacing="0.2em 0.2em 0.2em"> <mtr> <mtd> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> </mtd> <mtd> <mo>=</mo> <mfrac> <mi>λ</mi> <msup> <mi>u</mi> <mi>γ</mi> </msup> </mfrac> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mtext>in </mtext> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> </mtd> <mtd> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> </mtd> <mtd> <mtext>on </mtext> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> <EquationSource Format="TEX"> $\begin{aligned} \left \{ \begin{aligned} -\Delta u &=\frac{\lambda}{u^{\gamma}}+f(x,u) && \text{in }\Omega , \\ u &=0 \qquad &&\text{on }\partial \Omega , \end{aligned} \right . \end{aligned}$ </EquationSource> </Equation> where Δ denotes the discrete Laplacian, V denotes the vertices set and E denotes the edges set, <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mn>0</mn> <mo><</mo> <mi>γ</mi> <mo><</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$0<\gamma <1$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <mi>V</mi> </math></EquationSource> <EquationSource Format="TEX">$\Omega \subset V$</EquationSource> </InlineEquation> is a bounded domain and <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>λ</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\lambda >0$</EquationSource> </InlineEquation>. By employing a local minimization approach within the variational framework, we establish the existence of a strictly positive solution to the above problem, when f has a subquadratic growth.

Existence result for an elliptic equation on locally finite graphs with singularity

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2026年2月9日

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Chenyi Yan,
Junping Xie,
Xingyong Zhang,
Xuechen Zhang

摘要

We study the following equation with singular nonlinearity on locally finite graphs $G = (V, E)$ $G=(V,E)$ : ${\begin{aligned} - Δ u & = \frac{λ}{u^{γ}} + f (x, u) & in Ω, \\ u & = 0 & on \partial Ω, \end{aligned}$ $\begin{aligned} \left \{ \begin{aligned} -\Delta u &=\frac{\lambda}{u^{\gamma}}+f(x,u) && \text{in }\Omega , \\ u &=0 \qquad &&\text{on }\partial \Omega , \end{aligned} \right . \end{aligned}$ where Δ denotes the discrete Laplacian, V denotes the vertices set and E denotes the edges set, $0 < γ < 1$ $0<\gamma <1$ , $Ω \subset V$ $\Omega \subset V$ is a bounded domain and $λ > 0$ $\lambda >0$ . By employing a local minimization approach within the variational framework, we establish the existence of a strictly positive solution to the above problem, when f has a subquadratic growth.