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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2504.19817 |
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| _version_ | 1866916710191202304 |
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| author | He, Qihan Liu, Wenxuan Pan, Yiqing |
| author_facet | He, Qihan Liu, Wenxuan Pan, Yiqing |
| contents | We consider the existence, non-existence and multiplicity of positive solutions to the following critical Hardy-Hénon equation with logarithmic term \begin{equation*}\label{eq11}\left\{ \begin{array}{ll} -Δu =|x|^α|u|^{2^*_α-2}\cdot u+μu\log u^2+λu, &x\in Ω,\\ u=0, &x\in \partial Ω,\\ \end{array} \right.\end{equation*} where $ Ω=B$ for $α\geq 0$, $ Ω=B\setminus\{0\}$ for $α\in(-2,0)$, $B\subset\mathbb{R}^N$ is an unit ball, $λ, μ\in \mathbb{R}$, $N\geq 3, α>-2$, $2^*_α:=\frac{2(N+α)}{N-2}$ is the critical exponent for the embedding $H_{0,r}^{1}( Ω)\hookrightarrow L^p( Ω;|x|^α)$, and which can be seen as a Brézis-Nirenberg problem. When $N \geq 4$ and $μ>0$, we will show that the above problem has a positive Mountain pass solution, which is also a ground state solution. At the same time, when $μ<0$, under some assumptions on the $N$, $μ$, $λ$ and $α$, we will show that the above problem has at least a positive least energy solution and at least a positive Mountain pass solution, respectively. What's more, when certain inequality related to $N \geq 3$, $μ<0 $ and $α\in(-2,0]$ holds, we will demonstrate the non-existence of positive solutions to the above-mentioned problem. The presence of logarithmic term brings some new and interesting phenomena to this problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19817 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positive solutions of critical Hardy-Hénon equations with logarithmic term He, Qihan Liu, Wenxuan Pan, Yiqing Analysis of PDEs We consider the existence, non-existence and multiplicity of positive solutions to the following critical Hardy-Hénon equation with logarithmic term \begin{equation*}\label{eq11}\left\{ \begin{array}{ll} -Δu =|x|^α|u|^{2^*_α-2}\cdot u+μu\log u^2+λu, &x\in Ω,\\ u=0, &x\in \partial Ω,\\ \end{array} \right.\end{equation*} where $ Ω=B$ for $α\geq 0$, $ Ω=B\setminus\{0\}$ for $α\in(-2,0)$, $B\subset\mathbb{R}^N$ is an unit ball, $λ, μ\in \mathbb{R}$, $N\geq 3, α>-2$, $2^*_α:=\frac{2(N+α)}{N-2}$ is the critical exponent for the embedding $H_{0,r}^{1}( Ω)\hookrightarrow L^p( Ω;|x|^α)$, and which can be seen as a Brézis-Nirenberg problem. When $N \geq 4$ and $μ>0$, we will show that the above problem has a positive Mountain pass solution, which is also a ground state solution. At the same time, when $μ<0$, under some assumptions on the $N$, $μ$, $λ$ and $α$, we will show that the above problem has at least a positive least energy solution and at least a positive Mountain pass solution, respectively. What's more, when certain inequality related to $N \geq 3$, $μ<0 $ and $α\in(-2,0]$ holds, we will demonstrate the non-existence of positive solutions to the above-mentioned problem. The presence of logarithmic term brings some new and interesting phenomena to this problem. |
| title | Positive solutions of critical Hardy-Hénon equations with logarithmic term |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.19817 |