Sharp multiscale control for high order nonlinear equations
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911144476672000 |
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| author | Robert, Frédéric |
| author_facet | Robert, Frédéric |
| contents | We analyze the behavior of families $(u_α)_{α>0}$ of solutions to the high-order critical equation $P_αu_α=Δ_g^k u_α+\hbox{lot}=|u_α|^{2^\star-2}u_α$ on a Riemannian manifold $M$, with a uniform bound on the Dirichlet energy. We prove a sharp pointwise control of the $u_α$'s by a sum of bubbles uniformly with respect to $α\to +\infty$, that is $|u_α|\leq C\Vert u_\infty \Vert_\infty +C\sum_{i=1}^NB_{i,α}$ where $u_\infty \in C^{2k}(M)$ and the $(B_{i,α})_α$, $i=1,...,N$ are explicit standard peaks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_07517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp multiscale control for high order nonlinear equations Robert, Frédéric Analysis of PDEs Primary 35J35, Secondary 35J60, 35B44, 35J08, 58J05 We analyze the behavior of families $(u_α)_{α>0}$ of solutions to the high-order critical equation $P_αu_α=Δ_g^k u_α+\hbox{lot}=|u_α|^{2^\star-2}u_α$ on a Riemannian manifold $M$, with a uniform bound on the Dirichlet energy. We prove a sharp pointwise control of the $u_α$'s by a sum of bubbles uniformly with respect to $α\to +\infty$, that is $|u_α|\leq C\Vert u_\infty \Vert_\infty +C\sum_{i=1}^NB_{i,α}$ where $u_\infty \in C^{2k}(M)$ and the $(B_{i,α})_α$, $i=1,...,N$ are explicit standard peaks. |
| title | Sharp multiscale control for high order nonlinear equations |
| topic | Analysis of PDEs Primary 35J35, Secondary 35J60, 35B44, 35J08, 58J05 |
| url | https://arxiv.org/abs/2509.07517 |