Quantization effects for a fourth order equation of exponential growth in dimension four

We investigate the asymptotic behavior as $k\to +\mathrm{\infty }$ of sequences $(u_k{)}_{k\in \mathbb{N}}\in C^4(\mathrm{\Omega })$ of solutions of the equations $\Delta^2 u_k=V_k e^{4u_k}$on$\Omega$,where$\Omega$isaboundeddomainof$$\mathbb{R}$^4$and$\lim_{k\to +\infty}V_k=1$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 $\mathrm{\Delta }{u}_k$, we recover such a quantization as in dimension 2.