Multi-Bubble Blow-up Analysis for an Almost Critical Problem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911114402463744 |
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| author | Ayed, Mohamed Ben Mehdi, Khalil El |
| author_facet | Ayed, Mohamed Ben Mehdi, Khalil El |
| contents | Consider a smooth, bounded domain $Ø\subset \mathbb{R}^n$ with $n\geq 4$ and a smooth positive function $V$. We analyze the asymptotic behavior of a sequence of positive solutions $u_\e$ to the equation $-Δu +V(x)u =u^{\frac{n+2}{n-2}-\e}$ in $Ø$ with zero Dirichlet boundary conditions, as $\e\to 0$. We determine the precise blow-up rate and characterize the locations of interior concentration points in the general case of multiple blow-up, providing an exhaustive description of interior blow-up phenomena of this equation. Our result is established through a delicate analysis of the gradient of the corresponding Euler-Lagrange functional. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_17942 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multi-Bubble Blow-up Analysis for an Almost Critical Problem Ayed, Mohamed Ben Mehdi, Khalil El Analysis of PDEs 35A15, 35J20, 35J25 Consider a smooth, bounded domain $Ø\subset \mathbb{R}^n$ with $n\geq 4$ and a smooth positive function $V$. We analyze the asymptotic behavior of a sequence of positive solutions $u_\e$ to the equation $-Δu +V(x)u =u^{\frac{n+2}{n-2}-\e}$ in $Ø$ with zero Dirichlet boundary conditions, as $\e\to 0$. We determine the precise blow-up rate and characterize the locations of interior concentration points in the general case of multiple blow-up, providing an exhaustive description of interior blow-up phenomena of this equation. Our result is established through a delicate analysis of the gradient of the corresponding Euler-Lagrange functional. |
| title | Multi-Bubble Blow-up Analysis for an Almost Critical Problem |
| topic | Analysis of PDEs 35A15, 35J20, 35J25 |
| url | https://arxiv.org/abs/2502.17942 |