<p>In this paper, we consider the logistic elliptic equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2161_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="108" /> </InlineMediaObject> <EquationSource Format="TEX">$-\Delta u = u- u^{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>u</mi> <mo>-</mo> <msup> <mi>u</mi> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> in a smooth bounded domain <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2161_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \subset {\mathbb {R}}^{N},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2161_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$N\ge 2,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>2</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> equipped with the sublinear Neumann boundary condition <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2161_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{\partial u}{\partial \nu } = \mu u^{q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mrow> <mi>∂</mi> <mi>u</mi> </mrow> <mrow> <mi>∂</mi> <mi>ν</mi> </mrow> </mfrac> <mo>=</mo> <mi>μ</mi> <msup> <mi>u</mi> <mi>q</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2161_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$\partial \Omega ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2161_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">$0<q<1<p,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2161_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a parameter. With sub- and supersolutions and a comparison principle for the equation, we analyze the asymptotic profile of the unique positive solution for the equation as <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2161_Article_IEq8.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu \rightarrow \infty .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo stretchy="false">→</mo> <mi>∞</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

Boundary layer profiles of positive solutions for logistic equations with sublinear nonlinearity on the boundary

期刊论文
发表日期：

2025年8月8日

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Kenichiro Umezu

摘要

In this paper, we consider the logistic elliptic equation $-\Delta u = u- u^{p}$ $- Δ u = u - u^{p}$ in a smooth bounded domain $\Omega \subset {\mathbb {R}}^{N},$ $Ω \subset R^{N},$ $N\ge 2,$ $N \geq 2,$ equipped with the sublinear Neumann boundary condition $\frac{\partial u}{\partial \nu } = \mu u^{q}$ $\frac{\partial u}{\partial ν} = μ u^{q}$ on $\partial \Omega ,$ $\partial Ω,$ where $0<q<1<p,$ $0 < q < 1 < p,$ and $\mu \ge 0$ $μ \geq 0$ is a parameter. With sub- and supersolutions and a comparison principle for the equation, we analyze the asymptotic profile of the unique positive solution for the equation as $\mu \rightarrow \infty .$ $μ \to \infty .$