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

Abstract

We let $Ω$ be a smooth bounded domain of $\mathbb{R}^4$ and a sequence of fonctions $(V_k)_{k\in\mathbb{N}}\in C^0(Ω)$ such that $\lim_{k\to +\infty}V_k=1$ in $C^0_{loc}(Ω)$. We consider a sequence of functions $(u_k)_{k\in\mathbb{N}}\in C^4(Ω)$ such that $$Δ^2 u_k=V_k e^{4u_k}$$ in $Ω$ for all $k\in\mathbb{N}$. We address in this paper the question of the asymptotic behaviour of the $(u_k)'s$ when $k\to +\infty$. 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 $u_k$'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.