Blow-up phenomena for the constant Q/R-curvature equation
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910157453131776 |
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| author | Li, Caiyan Wang, Guofang Wei, Wei |
| author_facet | Li, Caiyan Wang, Guofang Wei, Wei |
| contents | Let $n\ge 25$ be an integer. In this paper, we construct a smooth metric $g_{0}$ on $\mathbb{S}^n$ with the property that the set of metrics in the conformal class of $g_{0}$ having positive scalar curvature and positive constant quotient $Q/R$ is non-compact. Equivalently, we construct families of solutions exhibiting blow-up behavior
for the following equation \begin{align*}
P _{g_{0}}u- \frac{ (n+2 )(n-4 )}{4} u^{ \frac{2}{n-4}} L_{g_{0}}u^{ \frac{n-2}{n-4}} =0, \quad u>0\quad\text{on} \ \mathbb{S}^{n}, \end{align*}
where $P _{g_{0}}$ is the Paneitz operator and $ L_{g_{0}}=-Δ_{g_{0}} +\frac{n-2}{4(n-1 )}R_{g_{0}} $ is the conformal Laplacian of $ g_{0}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20571 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Blow-up phenomena for the constant Q/R-curvature equation Li, Caiyan Wang, Guofang Wei, Wei Differential Geometry Analysis of PDEs Let $n\ge 25$ be an integer. In this paper, we construct a smooth metric $g_{0}$ on $\mathbb{S}^n$ with the property that the set of metrics in the conformal class of $g_{0}$ having positive scalar curvature and positive constant quotient $Q/R$ is non-compact. Equivalently, we construct families of solutions exhibiting blow-up behavior for the following equation \begin{align*} P _{g_{0}}u- \frac{ (n+2 )(n-4 )}{4} u^{ \frac{2}{n-4}} L_{g_{0}}u^{ \frac{n-2}{n-4}} =0, \quad u>0\quad\text{on} \ \mathbb{S}^{n}, \end{align*} where $P _{g_{0}}$ is the Paneitz operator and $ L_{g_{0}}=-Δ_{g_{0}} +\frac{n-2}{4(n-1 )}R_{g_{0}} $ is the conformal Laplacian of $ g_{0}$. |
| title | Blow-up phenomena for the constant Q/R-curvature equation |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2604.20571 |