Concentration phenomena for a fourth-order equation with exponential growth: The radial case |
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Authors: | Frédéric Robert |
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Institution: | Université de Nice-Sophia Antipolis, Laboratoire J.A. Dieudonné, Parc Valrose, 06108 Nice cedex 2, France |
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Abstract: | We let Ω be a smooth bounded domain of R4 and a sequence of functions (Vk)k∈N∈C0(Ω) such that limk→+∞Vk=1 in . We consider a sequence of functions (uk)k∈N∈C4(Ω) such that Δ2uk=Vke4uk in Ω for all k∈N. We address in this paper the question of the asymptotic behavior of the (uk)'s when k→+∞. The corresponding problem in dimension 2 was considered by Brézis and Merle, and Li and 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 in Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liouville equations in dimension four, J. Eur. Math. Soc., in press, available on http://www-math.unice.fr/~frobert], a similar quantization phenomenon does not hold for this fourth-order problem. Assuming that the uk's are radially symmetrical, we push further the analysis of the mentioned work. 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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Keywords: | primary 35J35 secondary 35B40 |
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