A Nonlinear PDE with Two Hardy–Sobolev Critical Exponents with One-Dimensional Singularity
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 the existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the equation\begin{equation*}-\Delta u+hu=\lambda\rho^{-s_1}_\Gamma u^{2^*_{s_1}-1}+\rho^{-s_2}_...
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| Datum: | 2026 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України
2026
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| Online Zugang: | https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1129 |
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| Назва журналу: | Journal of Mathematical Physics, Analysis, Geometry |
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Journal of Mathematical Physics, Analysis, Geometry| Zusammenfassung: | 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 the existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the equation\begin{equation*}-\Delta u+hu=\lambda\rho^{-s_1}_\Gamma u^{2^*_{s_1}-1}+\rho^{-s_2}_\Gamma u^{2^*_{s_2}-1} \quad \textrm{ in } \Omega, \tag{1}\end{equation*}where $h : \Omega \longrightarrow \mathbb{R}$ is a continuous function, $\lambda$ is a positive real parameter, $0\leq s_2<s_1<2$, and $\rho_\Gamma$ is the distance function to $\Gamma$. In this paper, we prove the existence of mountain pass solutions for the Euler-Lagrange equation (1) depending on the local geometry of the curve and the potential $h$. We also study the existence, symmetry and decay estimates of the positive entire solutions of (1) with $\Omega=\mathbb{R}^N$ and $\Gamma$ being the real line.
Mathematical Subject Classification 2020: 35J60, 35B33, 35A15, 35R45, 35B40 |
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