Normal conformal metrics with prescribed $Q$-Curvature in $\mathbb{R}^{2n}$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916627425001472 |
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| author | Huang, Xia Ye, Dong Zhou, Feng |
| author_facet | Huang, Xia Ye, Dong Zhou, Feng |
| contents | We consider the $Q$-curvature equation \begin{equation}\label{0.1} (-Δ)^n u = K(x)e^{2nu}\quad\text{in} ~\mathbb{R}^{2n} \ (n \geq 2) \end{equation} where $K$ is a given non constant continuous function. Under mild growth control on $K$, we get a necessary condition on the total curvature $Λ_u$ for any normal conformal metric $g_u = e^{2u}|dx|^2$ satisfying $Q_{g_u} = K$ in $\mathbb{R}^{2n}$, or equivalently, solutions to equation with logarithmic growth at infinity. Inversely, when $K$ is nonpositive satisfying polynomial growth control, we show the existence of normal conformal metrics with quasi optimal range of total curvature and precise asymptotic behavior at infinity. If furthermore $K$ is radial symmetric, we establish the same existence result without any growth assumption on $K$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_16881 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normal conformal metrics with prescribed $Q$-Curvature in $\mathbb{R}^{2n}$ Huang, Xia Ye, Dong Zhou, Feng Analysis of PDEs We consider the $Q$-curvature equation \begin{equation}\label{0.1} (-Δ)^n u = K(x)e^{2nu}\quad\text{in} ~\mathbb{R}^{2n} \ (n \geq 2) \end{equation} where $K$ is a given non constant continuous function. Under mild growth control on $K$, we get a necessary condition on the total curvature $Λ_u$ for any normal conformal metric $g_u = e^{2u}|dx|^2$ satisfying $Q_{g_u} = K$ in $\mathbb{R}^{2n}$, or equivalently, solutions to equation with logarithmic growth at infinity. Inversely, when $K$ is nonpositive satisfying polynomial growth control, we show the existence of normal conformal metrics with quasi optimal range of total curvature and precise asymptotic behavior at infinity. If furthermore $K$ is radial symmetric, we establish the same existence result without any growth assumption on $K$. |
| title | Normal conformal metrics with prescribed $Q$-Curvature in $\mathbb{R}^{2n}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.16881 |