<p>This paper is divided into two parts. First, we will prove the existence of solutions of the <i>p</i>-Laplacian equation in the Riemannian manifold in the space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1031_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {H}}^{\alpha ,p}_{loc}({\mathcal {N}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mrow> <mi mathvariant="italic">loc</mi> </mrow> <mrow> <mi>α</mi> <mo>,</mo> <mi>p</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. On the other hand, we will give a criterion to obtain a positive lower bound for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1031_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _{1,p}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where is a bounded domain <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1031_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \subset {\mathcal {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <mi mathvariant="script">N</mi> </mrow> </math></EquationSource> </InlineEquation>. In the first result, we do not consider a bounded subset on the Riemannian manifold <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1031_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation>. </p>

p-Laplacian problem in a Riemannian manifold

期刊论文
发表日期：

2025年2月26日

查看全文

J. Vanterler da C. Sousa,
Lamine Mbarki,
Leandro S. Tavares

摘要

This paper is divided into two parts. First, we will prove the existence of solutions of the p-Laplacian equation in the Riemannian manifold in the space ${\mathcal {H}}^{\alpha ,p}_{loc}({\mathcal {N}})$ $H_{loc}^{α, p} (N)$ . On the other hand, we will give a criterion to obtain a positive lower bound for $\lambda _{1,p}(\Omega )$ $λ_{1, p} (Ω)$ , where is a bounded domain $\Omega \subset {\mathcal {N}}$ $Ω \subset N$ . In the first result, we do not consider a bounded subset on the Riemannian manifold ${\mathcal {N}}$ $N$ .