Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (P-epsilon) : Delta(2)u = u(n+4/n-4) + epsilon u, u > 0 in Omega, u = Delta u = 0 on partial derivative Omega where Omega is a bounded and smooth domain in R-n, n > 8 and epsilon > 0. We analyze the asymptotic behavior of solutions of (P-epsilon) which are minimizing for the Sobolev inequality as epsilon -> 0 and we prove existence of solutions to (P-epsilon) which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for epsilon small, (P-epsilon) has at least as many solutions as the Ljustemik-Schnirelman category of Omega. (c) 2005 Elsevier Ltd. All rights reserved.