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Concentration phenomena for a fourth order equations with exponential growth: the radial case | | Frederic Robert
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7 Dec 2005 | | Subject: | Analysis of PDEs | | Abstract: | We let $Omega$ be a smooth bounded domain of $mathbb{R}^4$ and a sequence of fonctions $(V_k)_{kinmathbb{N}}in C^0(Omega)$ such that $lim_{k o +infty}V_k=1$ in $C^0_{loc}(Omega)$. We consider a sequence of functions $(u_k)_{kinmathbb{N}}in C^4(Omega)$ such that $$Delta^2 u_k=V_k e^{4u_k}$$ in $Omega$ for all $kinmathbb{N}$. We address in this paper the question of the asymptotic behaviour of the $(u_k)’s$ when $k o +infty$. 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. | | Source: | arXiv, math/0512149 | | Services: | Forum | Review | PDF | Favorites | |
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