Variational problems related to self-similar solutions of Hardy-Sobolev heat equation in RN
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866916045165428736 |
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| author | Fang, Fei Tan, Zhong |
| author_facet | Fang, Fei Tan, Zhong |
| contents | In this paper, we apply a self-similar transformation to convert the parabolic equation with a Sobolev-Hardy term \begin{align*} u_t-Δu= \frac{|u|^{q-2}u}{\left|x\right|^s} & \text { in } \mathbb{R}^N \times(0, \infty), \end{align*} into the following elliptic equation \begin{equation*} -Δv-\frac{1}{2} y \cdot \nabla v=αv+ \frac{|v|^{q-2} v}{|y|^s}, \end{equation*} where $2 < q \leq 2^*(s)=\frac{2 N-2 s}{N-2}, 0 \leq s < 2, α=\frac{2-s}{2q-4}$. For this equation, we establish the weighted Hardy inequality and Sobolev inequality. Furthermore, by virtue of the variational methods, we obtain infinitely many solutions in the subcritical case, and prove the existence of solutions in the critical case. We also apply the Pohozaev identity to establish the nonexistence of solutions under certain conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24804 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Variational problems related to self-similar solutions of Hardy-Sobolev heat equation in RN Fang, Fei Tan, Zhong Analysis of PDEs In this paper, we apply a self-similar transformation to convert the parabolic equation with a Sobolev-Hardy term \begin{align*} u_t-Δu= \frac{|u|^{q-2}u}{\left|x\right|^s} & \text { in } \mathbb{R}^N \times(0, \infty), \end{align*} into the following elliptic equation \begin{equation*} -Δv-\frac{1}{2} y \cdot \nabla v=αv+ \frac{|v|^{q-2} v}{|y|^s}, \end{equation*} where $2 < q \leq 2^*(s)=\frac{2 N-2 s}{N-2}, 0 \leq s < 2, α=\frac{2-s}{2q-4}$. For this equation, we establish the weighted Hardy inequality and Sobolev inequality. Furthermore, by virtue of the variational methods, we obtain infinitely many solutions in the subcritical case, and prove the existence of solutions in the critical case. We also apply the Pohozaev identity to establish the nonexistence of solutions under certain conditions. |
| title | Variational problems related to self-similar solutions of Hardy-Sobolev heat equation in RN |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.24804 |