BLOWING UP AND MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER EQUATION WITH CRITICAL NONLINEARITY

In this paper, we consider the following nonlinear elliptic problem : Delta(2)u = vertical bar u vertical bar(8/n-4)u + mu vertical bar u vertical bar(q-1)u, in Omega, Delta u = u = 0 on partial derivative Omega, where Omega is a bounded and smooth domain in R-n, n is an element of {5,6, 7}, mu is a parameter and q is an element of]4/(n - 4), (12 - n)/(n - 4)[. We study the solutions which concentrate around two points of Omega. We prove that the concentration speeds are the same order and the distances of the concentration points from each other and from the boundary are bounded. For Omega = (Omega(a))(a) a smooth ringshaped open set, we establish the existence of positive solutions which concentrate at two points of Omega. Finally, we show that for mu > 0, large enough, the problem has at least many positive solutions as the Ljusternik-Schnirelman category of Omega.