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

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