<p>This paper investigates a class of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation>-Laplacian singular differential equations with an indefinite weighted parameter <Equation ID="Equ68"> <EquationSource Format="TEX">$\begin{aligned} (\phi (x'))'+\frac{g(t)}{x^\rho }=h(t)x^{\delta }+s\cdot e(t), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>x</mi> <mi>ρ</mi> </msup> </mfrac> <mo>=</mo> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>x</mi> <mi>δ</mi> </msup> <mo>+</mo> <mi>s</mi> <mo>·</mo> <mi>e</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$e\in C(\mathbb {R}/T\mathbb {Z};\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>e</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">/</mo> <mi>T</mi> <mi mathvariant="double-struck">Z</mi> <mo>;</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is sign-changing, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$s\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is a parameter, and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$T>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. By employing the method of upper and lower solutions and Leray-Schauder degree, we establish an Ambrosetti-Prodi type result for the equation in the cases where the weight function <i>e</i> is either strictly positive or sign-changing. In addition, we analyze the asymptotic behavior of the periodic solutions as the parameter tends to infinity.</p>

Ambrosetti-Prodi periodic problem for $\phi $-Laplacian singular differential equations with an indefinite weighted parameter

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发表日期：

2026年5月20日

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Yun Xin,
Qigang Yuan

摘要

This paper investigates a class of $\phi $ $ϕ$ -Laplacian singular differential equations with an indefinite weighted parameter $\begin{aligned} (\phi (x'))'+\frac{g(t)}{x^\rho }=h(t)x^{\delta }+s\cdot e(t), \end{aligned}$ $\begin{matrix} {(ϕ (x^{'}))}^{'} + \frac{g (t)}{x^{ρ}} = h (t) x^{δ} + s \cdot e (t), \end{matrix}$ where $e\in C(\mathbb {R}/T\mathbb {Z};\mathbb {R})$ $e \in C (R / T Z; R)$ is sign-changing, $s\in \mathbb {R}$ $s \in R$ is a parameter, and $T>0$ $T > 0$ . By employing the method of upper and lower solutions and Leray-Schauder degree, we establish an Ambrosetti-Prodi type result for the equation in the cases where the weight function e is either strictly positive or sign-changing. In addition, we analyze the asymptotic behavior of the periodic solutions as the parameter tends to infinity.

Ambrosetti-Prodi periodic problem for \(\phi \)-Laplacian singular differential equations with an indefinite weighted parameter

摘要