Quantization Effects for a Fourth Order Equation of Exponential Growth in Dimension Four

We investigate the asymptotic behavior as k → +∞ of sequences (uk)k∈N ∈ C (Ω) of solutions of the equations ∆uk = Vke 4uk on Ω, where Ω is a bounded domain of R and limk→+∞ Vk = 1 in C 0 loc (Ω). The corresponding 2-dimensional problem was studied by Brézis-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 [1], such a quantization does not hold in dimension four for the problem in its full generality. We prove here that under natural hypothesis on ∆uk, we recover such a quantization as in dimension 2.