Xiyou Cheng, Huijuan Shao, Lei Wei, Meihua Yang
Abstract:
We consider the elliptic equation
$$
-\Delta u=d(x)^\alpha u^p,\quad x\in \Omega\quad (\text{or }
x\in\mathbb{R}^N\setminus \overline{\Omega}),
$$
where \(\alpha, p \in \mathbb{R}\), \(d(x) = \text{dist} (x,\partial\Omega)\) and
\(\Omega\subset\mathbb{R}^N\) \((N\geq 3)\) is a bounded smooth domain.
We establish estimates for the positive solutions when
\(1< p< \frac{N+2}{N-2}\),
and the nonexistence of positive solutions for exterior domains.
For the corresponding Dirichlet problem we show the existence and
uniqueness of positive solutions.
Submitted April 27, 2026. Published September 22, 2026.
Math Subject Classifications: 35J61, 35A09, 35B09, 35B45.
Key Words: Hardy-Henon equations; existence; uniqueness.
DOI: 10.58997/ejde.2026.72
Show me the PDF file (411 KB), TEX file for this article.
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Xiyou Cheng School of Mathematics and Statistics Lanzhou University Lanzhou, Gansu 730000, China email: chengxy@lzu.edu.cn |
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Huijuan Shao School of Mathematics and Statistics Jiangsu Normal University Xuzhou, Jiangsu 221116, China email: 2859192413@qq.com |
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Lei Wei School of Mathematics and Statistics Jiangsu Normal University Xuzhou, Jiangsu 221116, China email: wlxznu@163.com |
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Meihua Yang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan, Hubei 430074, China email: yangmeih@hust.edu.cn |
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