Influence of the curvature in the existence of solutions for a two Hardy-Sobolev critical exponents
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914848934199296 |
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| author | Thiam, El Hadji Abdoulaye Diatta, Abdourahmane |
| author_facet | Thiam, El Hadji Abdoulaye Diatta, Abdourahmane |
| contents | For $N\geq 4$, we let $Ω$ be a bounded domain of $\mathbb{R}^N$ and $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the equation \begin{equation}\label{Atusi} -Δu+hu=λρ^{-s_1}_Γu^{2^*_{s_1}-1}+ρ^{-s_2}_Γu^{2^*_{s_2}-1} \qquad \textrm{ in } Ω\end{equation} where $h$ is a continuous function and $ρ_Γ$ is the distance function to $Γ$. We prove the existence of a mountain pass solution for this Euler-Lagrange equation depending on the local geometry of the curve and the potential $h$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_04768 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Influence of the curvature in the existence of solutions for a two Hardy-Sobolev critical exponents Thiam, El Hadji Abdoulaye Diatta, Abdourahmane Analysis of PDEs For $N\geq 4$, we let $Ω$ be a bounded domain of $\mathbb{R}^N$ and $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the equation \begin{equation}\label{Atusi} -Δu+hu=λρ^{-s_1}_Γu^{2^*_{s_1}-1}+ρ^{-s_2}_Γu^{2^*_{s_2}-1} \qquad \textrm{ in } Ω\end{equation} where $h$ is a continuous function and $ρ_Γ$ is the distance function to $Γ$. We prove the existence of a mountain pass solution for this Euler-Lagrange equation depending on the local geometry of the curve and the potential $h$. |
| title | Influence of the curvature in the existence of solutions for a two Hardy-Sobolev critical exponents |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2309.04768 |