Blowing-up solutions to competitive critical systems in dimension 3
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911237802033152 |
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| author | Fernández, Antonio J. Medina, María Pistoia, Angela |
| author_facet | Fernández, Antonio J. Medina, María Pistoia, Angela |
| contents | We study the critical system of $m\geq 2$ equations \begin{equation*} -Δu_i = u_i^5 + \sum_{j = 1,\,j\neq i}^m β_{ij} u_i^2 u_j^3\,, \quad u_i \gneqq 0 \quad \mbox{in } \mathbb{R}^3\,, \quad i \in \{1, \ldots, m\}\,, \end{equation*} where $β_{κ\ell} =α\in\mathbb{R}$ if $κ\neq\ell$, and $β_{\ell m}=β_{m κ} =β<0$, for $ κ, \ell \in \{1,\ldots, m-1\}$. We construct solutions to this system in the case where $β\to-\infty$ by means of a Ljapunov-Schmidt reduction argument. This allows us to identify the explicit form of the solution at main order: $u_1$ will look like a perturbation of the standard radial positive solution to the Yamabe equation, while $u_2$ will blow-up at the $k$ vertices of a regular planar polygon. The solutions to the other equations will replicate the blowing-up structure under an appropriate rotation that ensures $u_i\neq u_j$ for $i\neq j$. The result provides the first almost-explicit example of non-synchronized solutions to competitive critical systems in dimension 3. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_07951 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Blowing-up solutions to competitive critical systems in dimension 3 Fernández, Antonio J. Medina, María Pistoia, Angela Analysis of PDEs 35J47, 35B33 We study the critical system of $m\geq 2$ equations \begin{equation*} -Δu_i = u_i^5 + \sum_{j = 1,\,j\neq i}^m β_{ij} u_i^2 u_j^3\,, \quad u_i \gneqq 0 \quad \mbox{in } \mathbb{R}^3\,, \quad i \in \{1, \ldots, m\}\,, \end{equation*} where $β_{κ\ell} =α\in\mathbb{R}$ if $κ\neq\ell$, and $β_{\ell m}=β_{m κ} =β<0$, for $ κ, \ell \in \{1,\ldots, m-1\}$. We construct solutions to this system in the case where $β\to-\infty$ by means of a Ljapunov-Schmidt reduction argument. This allows us to identify the explicit form of the solution at main order: $u_1$ will look like a perturbation of the standard radial positive solution to the Yamabe equation, while $u_2$ will blow-up at the $k$ vertices of a regular planar polygon. The solutions to the other equations will replicate the blowing-up structure under an appropriate rotation that ensures $u_i\neq u_j$ for $i\neq j$. The result provides the first almost-explicit example of non-synchronized solutions to competitive critical systems in dimension 3. |
| title | Blowing-up solutions to competitive critical systems in dimension 3 |
| topic | Analysis of PDEs 35J47, 35B33 |
| url | https://arxiv.org/abs/2411.07951 |