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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.13239 |
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| _version_ | 1866917015518707712 |
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| author | Li, Tuoxin Wei, Juncheng Yang, Haidong |
| author_facet | Li, Tuoxin Wei, Juncheng Yang, Haidong |
| contents | In this paper, we consider the following prescribed scalar curvature problem:
\begin{equation*}
-Δu = K(x) u^{\frac{n+2}{n-2}}, \quad u>0\quad\hbox{in}\quad \mathbb{R}^n, \quad
u \in D^{1,2}(\mathbb{R}^n),
\end{equation*}
where $K(x)$ is a volcano-like positive function such that
$$ K(x)= K(r_0)- c_0 | |x|- r_0|^m + O( | |x|- r_0|^{m+θ}),\quad r_0- δ<|x| <r_0+δ$$
with $K(r_0), c_0, δ>0, θ>2, \min \{\frac{n-2}{2}, 2\} < m< n-2$.
We first prove the existence of infinitely many positive solutions. A consequence of our proof yields that the infinitely many solutions constructed in \cite{WY} are non-degenerate in the whole $D^{1, 2}(\mathbb{R}^{n})$ space. To our knowledge, it seems to be the first result of infinitely many solutions of prescribed scalar curvature problem when the potential function $K(x)$ is not radial. Our non-degeneracy results are also more complete and improve the result in \cite{GMPS}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13239 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitely many solutions for the prescribed scalar curvature problem with volcano-like curvature Li, Tuoxin Wei, Juncheng Yang, Haidong Analysis of PDEs In this paper, we consider the following prescribed scalar curvature problem: \begin{equation*} -Δu = K(x) u^{\frac{n+2}{n-2}}, \quad u>0\quad\hbox{in}\quad \mathbb{R}^n, \quad u \in D^{1,2}(\mathbb{R}^n), \end{equation*} where $K(x)$ is a volcano-like positive function such that $$ K(x)= K(r_0)- c_0 | |x|- r_0|^m + O( | |x|- r_0|^{m+θ}),\quad r_0- δ<|x| <r_0+δ$$ with $K(r_0), c_0, δ>0, θ>2, \min \{\frac{n-2}{2}, 2\} < m< n-2$. We first prove the existence of infinitely many positive solutions. A consequence of our proof yields that the infinitely many solutions constructed in \cite{WY} are non-degenerate in the whole $D^{1, 2}(\mathbb{R}^{n})$ space. To our knowledge, it seems to be the first result of infinitely many solutions of prescribed scalar curvature problem when the potential function $K(x)$ is not radial. Our non-degeneracy results are also more complete and improve the result in \cite{GMPS}. |
| title | Infinitely many solutions for the prescribed scalar curvature problem with volcano-like curvature |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.13239 |