Concentration phenomena for a fourth order equations with exponential growth: the radial case

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We let Ω be a smooth bounded domain of R⁴ and a sequence of fonctions (Vₖ)k∈ C⁰(Ω) such that ₖ→ ₊∞Vₖ=1 in C⁰loc(Ω). We consider a sequence of functions (uₖ)k∈ C⁴(Ω) such that Δ² uₖ=Vₖ e⁴ᵘᵏ in Ω for all k. We address in this paper the question of the asymptotic behaviour of the (uₖ)'s when k→ +∞. The corresponding problem in dimension 2 was considered by Brézis-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 uₖ'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.