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Concentration phenomena for a fourth order equations with exponential growth: the radial case
Analysis of PDEs · Submitted 7 Dec 2005
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that there are exactly three types of blow-up and we describe each type in a very detailed way.
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We let $\Omega$ be a smooth bounded domain of $\mathbb{R}^4$ and a sequence of fonctions $(V_k)_{k\in\mathbb{N}}\in C^0(\Omega)$ such that $\lim_{k\to +\infty}V_k=1$ in $C^0_{loc}(\Omega)$. We consider a sequence of functions $(u_k)_{k\in\mathbb{N}}\in C^4(\Omega)$ such that $$\Delta^2 u_k=V_k e^{4u_k}$$ in $\Omega$ for all $k\in\mathbb{N}$. We address in this paper the question of the asymptotic behaviour of the $(u_k)'s$ when $k\to +\infty$. The corresponding problem in dimension 2 was considered by Br\'ezis-Merle and Li-Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author, a similar quantization phenomenon does not hold for this fourth order problem. Assuming that the $u_k$'s are radially symmetrical, we push further the previous analysis. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We let $\Omega$ be a smooth bounded domain of $\mathbb{R}^4$ and a sequence of fonctions $(V_k)_{k\in\mathbb{N}}\in C^0(\Omega)$ such that $\lim_{k\to +\infty}V_k=1$ in $C^0_{loc}(\Omega)$. | background 0.84 | 0.52 | 0.02 | asserted 0.97 |
| s1 | We consider a sequence of functions $(u_k)_{k\in\mathbb{N}}\in C^4(\Omega)$ such that $$\Delta^2 u_k=V_k e^{4u_k}$$ in $\Omega$ for all $k\in\mathbb{N}$. | background 0.60 | 0.60 | 0.02 | asserted 0.97 |
| s2 | We address in this paper the question of the asymptotic behaviour of the $(u_k)'s$ when $k\to +\infty$. | contribution 0.77 | 0.57 | 0.03 | asserted 0.50 |
| s3 | The corresponding problem in dimension 2 was considered by Br\'ezis-Merle and Li-Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. | background 1.00 | 0.03 | 0.03 | asserted 1.00 |
| s4 | Surprisingly, as shown by Adimurthi, Struwe and the author, a similar quantization phenomenon does not hold for this fourth order problem. | result 0.51 | 0.65 | 0.21 | asserted 1.00 |
| s5 | Assuming that the $u_k$'s are radially symmetrical, we push further the previous analysis. | contribution 0.56 | 0.72 | 0.04 | asserted 0.71 |
| s6 | We prove that there are exactly three types of blow-up and we describe each type in a very detailed way. | result 0.94 | 0.96 | 0.73 | asserted 1.00 |
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Original abstract
We let $\Omega$ be a smooth bounded domain of $\mathbb{R}^4$ and a sequence of fonctions $(V_k)_{k\in\mathbb{N}}\in C^0(\Omega)$ such that $\lim_{k\to +\infty}V_k=1$ in $C^0_{loc}(\Omega)$. We consider a sequence of functions $(u_k)_{k\in\mathbb{N}}\in C^4(\Omega)$ such that $$\Delta^2 u_k=V_k e^{4u_k}$$ in $\Omega$ for all $k\in\mathbb{N}$. We address in this paper the question of the asymptotic behaviour of the $(u_k)'s$ when $k\to +\infty$. The corresponding problem in dimension 2 was considered by Br\'ezis-Merle and Li-Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author, a similar quantization phenomenon does not hold for this fourth order problem. Assuming that the $u_k$'s are radially symmetrical, we push further the previous analysis. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way.
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