Abstract <p> Let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3014_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(p_1, p_2>1\)</EquationSource> </InlineEquation>, we consider the following sum of two different <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3014_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-Laplacians problem <Equation ID="Equi"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3014_Article_Equi.gif" Format="GIF" Height="85" Rendition="HTML" Resolution="72" Type="Linedraw" Width="296" /> </MediaObject> <EquationSource Format="TEX">\(\left\{ \begin{aligned} \, &-L_{p_1}u-L_{p_2}u=a(t)u^{\sigma}\quad \text{on}\ \ (0,1), \\ &\underset{t \longrightarrow 0}\lim A(t)\bigl(| u' |^{p_1-2}u' +|u'|^{p_2-2}u'\bigr) (t)=0, \\ &u(1)=0, \end{aligned} \right.\)</EquationSource> </Equation> where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3014_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="170" /> </InlineMediaObject> <EquationSource Format="TEX">\(0<\sigma<\min(p_1, p_2)-1\)</EquationSource> </InlineEquation> and the operator <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3014_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_{p}u\)</EquationSource> </InlineEquation> is defined by <Equation ID="Equii"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3014_Article_Equii.gif" Format="GIF" Height="36" Rendition="HTML" Resolution="72" Type="Linedraw" Width="158" /> </MediaObject> <EquationSource Format="TEX">\(L_{p}u:=\dfrac{1}{A}(A| u' |^{p-2} u')'\)</EquationSource> </Equation> for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3014_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(p>1\)</EquationSource> </InlineEquation>. We provide sufficient conditions on the functions <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3014_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(A\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3014_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(a\)</EquationSource> </InlineEquation> that yield the existence, and we give the asymptotic behavior of radial positive solutions. An example is given to illustrate the applicability of our main results. </p>

About the Radial Solutions of the Nonlinear \((p_1,p_2)\)-Laplacian Problem

期刊论文
发表日期：

2025年2月1日

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B. Khamessi

摘要

Abstract

Let \(p_1, p_2>1\) , we consider the following sum of two different \(p\) -Laplacians problem \(\left\{ \begin{aligned} \, &-L_{p_1}u-L_{p_2}u=a(t)u^{\sigma}\quad \text{on}\ \ (0,1), \\ &\underset{t \longrightarrow 0}\lim A(t)\bigl(| u' |^{p_1-2}u' +|u'|^{p_2-2}u'\bigr) (t)=0, \\ &u(1)=0, \end{aligned} \right.\) where \(0<\sigma<\min(p_1, p_2)-1\) and the operator \(L_{p}u\) is defined by \(L_{p}u:=\dfrac{1}{A}(A| u' |^{p-2} u')'\) for \(p>1\) . We provide sufficient conditions on the functions \(A\) and \(a\) that yield the existence, and we give the asymptotic behavior of radial positive solutions. An example is given to illustrate the applicability of our main results.