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

Abstract

We investigate the asymptotic behavior as $k \to +\infty$ of sequences $(u_k)_{k\in\mathbb{N}}\in C^4(Ω)$ of solutions of the equations $Δ^2 u_k=V_k e^{4u_k}$ on $Ω$, where $Ω$ is a bounded domain of $\mathbb{R}^4$ and $\lim_{k\to +\infty}V_k=1$ 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, such a quantization does not hold in dimension four for the problem in its full generality. We prove here that under natural hypothesis on $Δu_k$, we recover such a quantization as in dimension 2.