Concentration phenomena for a fourth-order equation with exponential growth: The radial case

We let Omega be a smooth bounded domain of R-4 and a sequence of functions (V-k)(k is an element of N) is an element of C-0 (Omega) such that lim(k ->+infinity) V-k = 1 in C-loc(0) (Omega). We consider a sequence of functions (u(k))(k is an element of N) is an element of C-4(Omega) such that Delta(2)u(k) = V(k)e(4uk) in Omega for all k is an element of N. We address in this paper the question of the asymptotic behavior of the (u(k))'s when k -> +infinity. The corresponding problem in dimension 2 was considered by Brezis and Merle, and Li and 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 in [Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Lionville equations in dimension four, J. Eur. Math. Soc., in press, available on http://www-math.unice.fr/-frobert], 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 analysis of the mentioned work. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way. (c) 2006 Elsevier Inc. All rights reserved.