RECORD
Influence of the curvature in the existence of solutions for a two Hardy-Sobolev critical exponents
Analysis of PDEs · Submitted 9 Sep 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $h$.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
For $N\geq 4$, we let $\Omega$ be a bounded domain of $\mathbb{R}^N$ and $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the equation \begin{equation}\label{Atusi} -\Delta u+hu=\lambda\rho^{-s_1}_\Gamma u^{2^*_{s_1}-1}+\rho^{-s_2}_\Gamma u^{2^*_{s_2}-1} \qquad \textrm{ in } \Omega \end{equation} where $h$ is a continuous function and $\rho_\Gamma$ is the distance function to $\Gamma$. 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 $h$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For $N\geq 4$, we let $\Omega$ be a bounded domain of $\mathbb{R}^N$ and $\Gamma$ be a closed curve contained in $\Omega$. | background 0.99 | 0.18 | 0.01 | asserted 0.87 |
| s1 | We study existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the equation \begin{equation}\label{Atusi} -\Delta u+hu=\lambda\rho^{-s_1}_\Gamma u^{2^*_{s_1}-1}+\rho^{-s_2}_\Gamma u^{2^*_{s_2}-1} \qquad \textrm{ in } \Omega \end{equation} where $h$ is a continuous function and $\rho_\Gamma$ is the distance function to $\Gamma$. | contribution 0.68 | 0.57 | 0.04 | asserted 0.82 |
| s2 | 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 $h$. | result 0.92 | 0.96 | 0.93 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
For $N\geq 4$, we let $\Omega$ be a bounded domain of $\mathbb{R}^N$ and $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the equation \begin{equation}\label{Atusi} -\Delta u+hu=\lambda\rho^{-s_1}_\Gamma u^{2^*_{s_1}-1}+\rho^{-s_2}_\Gamma u^{2^*_{s_2}-1} \qquad \textrm{ in } \Omega \end{equation} where $h$ is a continuous function and $\rho_\Gamma$ is the distance function to $\Gamma$. 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 $h$.
Human review
How the literature uses it
What this paper does with the arXiv papers it cites
What it cites
Cites as background 1 paper
-
For more results in this direction, see the works of Ghoussoub and Robert in [ref] , Demyanov and Nazarov [ref] , Chern and Lin [ref] , Lin and Li [ref] , the first author, Fall and Minlend in [ref] and the references there in.
On the Hardy-Schr\"odinger operator with a boundary singularity · Introduction · 0.91
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.