Asymptotic behavior of solutions to a planar Hartree equation with isolated singularities
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912873216737280 |
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| author | Feng, Tao Yang, Minbo Zhou, Xianmei |
| author_facet | Feng, Tao Yang, Minbo Zhou, Xianmei |
| contents | In this paper we investigate the isolated singularities of the Hartree type equation
\begin{equation*}
-Δu (x)= \left(\frac{1}{|x|^α}*e^u\right)e^{u(x)}\quad \text{in } B_{1}\setminus\{0\} ,
\end{equation*}
where $α>0$, $\displaystyle \frac{1}{|x|^α}*e^u\triangleq\int_{B_{1} \setminus \{0\}}\frac{e^u(y)}{|x-y|^α}dy$, and the punctured ball $B_{1}\setminus\{0\}\subset \mathbb{R}^2$. Under the finite total curvature condition, by establishing a representation formula for singular solutions, we obtain the asymptotic behavior of the solutions near the origin. We also extend this asymptotic behavior results to the case with a general non-negative coefficient $K(x)$, and to the higher-order Hartree-type equations in any dimension $n \geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03559 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of solutions to a planar Hartree equation with isolated singularities Feng, Tao Yang, Minbo Zhou, Xianmei Analysis of PDEs 35A21, 35B40, 35J91 In this paper we investigate the isolated singularities of the Hartree type equation \begin{equation*} -Δu (x)= \left(\frac{1}{|x|^α}*e^u\right)e^{u(x)}\quad \text{in } B_{1}\setminus\{0\} , \end{equation*} where $α>0$, $\displaystyle \frac{1}{|x|^α}*e^u\triangleq\int_{B_{1} \setminus \{0\}}\frac{e^u(y)}{|x-y|^α}dy$, and the punctured ball $B_{1}\setminus\{0\}\subset \mathbb{R}^2$. Under the finite total curvature condition, by establishing a representation formula for singular solutions, we obtain the asymptotic behavior of the solutions near the origin. We also extend this asymptotic behavior results to the case with a general non-negative coefficient $K(x)$, and to the higher-order Hartree-type equations in any dimension $n \geq 3$. |
| title | Asymptotic behavior of solutions to a planar Hartree equation with isolated singularities |
| topic | Analysis of PDEs 35A21, 35B40, 35J91 |
| url | https://arxiv.org/abs/2602.03559 |