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29 March 2024
 
  » arxiv » math/0512148

 Article overview


Quantization effects for a fourth order equation of exponential growth in dimension four
Frederic Robert ; PostScript ; PDF ; Other formats ;
Date 7 Dec 2005
Subject Analysis of PDEs
AbstractWe investigate the asymptotic behavior as $k o +infty$ of sequences $(u_k)_{kinmathbb{N}}in C^4(Omega)$ of solutions of the equations $Delta^2 u_k=V_k e^{4u_k}$ on $Omega$, where $Omega$ is a bounded domain of $mathbb{R}^4$ and $lim_{k o +infty}V_k=1$ in $C^0_{loc}(Omega)$. The corresponding 2-dimensional problem was studied by Br’ezis-Merle and Li-Shafrir who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Struwe and the author, such a quantization does not hold in dimension four for the problem in its full generality. We prove here that under natural hypothesis on $Delta u_k$, we recover such a quantization as in dimension 2.
Source arXiv, math/0512148
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