<p>In this paper we study the following quasilinear nonlocal differential equations with convolution coefficients <Equation ID="Equ27"> <EquationSource Format="TEX">$ -M\bigl ((a * u^\gamma )(1)\bigr ) (\varphi _p(u'))' = f(\lambda , x, u), \quad x \in (0,1), $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>-</mo> <mi>M</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mrow /> <mo>∗</mo> <msup> <mi>u</mi> <mi>γ</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gamma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varphi _p(u)=|u|^{p-2}u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>φ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is the <i>p</i>-Laplacian operator, and the reaction term <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f(\lambda , x, u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> includes linear, eigenvalue, sublinear, and logistic cases. We establish existence results for positive solutions and describe the global structure of the solution set in the degenerate case. The proofs of the main results are based upon fixed point arguments.</p>

Positive solutions for quasilinear non-local differential equations with convolution coefficients

期刊论文
发表日期：

2026年5月30日

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Ruyun Ma,
Xiangbing Lei

摘要

In this paper we study the following quasilinear nonlocal differential equations with convolution coefficients $ -M\bigl ((a * u^\gamma )(1)\bigr ) (\varphi _p(u'))' = f(\lambda , x, u), \quad x \in (0,1), $ $- M ((a * u^{γ}) (1)) {(φ_{p} (u^{'}))}^{'} = f (λ, x, u), x \in (0, 1),$ where $\gamma >0$ $γ > 0$ , $\varphi _p(u)=|u|^{p-2}u$ $φ_{p} (u) = {| u |}^{p - 2} u$ with $p>1$ $p > 1$ is the p-Laplacian operator, and the reaction term $f(\lambda , x, u)$ $f (λ, x, u)$ includes linear, eigenvalue, sublinear, and logistic cases. We establish existence results for positive solutions and describe the global structure of the solution set in the degenerate case. The proofs of the main results are based upon fixed point arguments.