Abstract
This paper is concerned with a biharmonic equation under the Navier boundary condition (P-/+epsilon): Delta(2)u = u(n+4/n-4-/+epsilon), u > 0 in Omega and u = Delta u = 0 on partial derivative Omega, where Omega is a smooth bounded domain in R-n, n >= 5, and epsilon > 0. We study the asymptotic behavior of solutions of (P-epsilon) which are minimizing for the Sobolev quotient as epsilon goes to zero. We show that such solutions concentrate around a point x(0) is an element of Omega as epsilon --> 0, moreover x(0) is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point x(0) of the Robin's function, there exist solutions of (P-epsilon) concentrating around x(0) as epsilon --> 0. Finally we prove that, in contrast with what happened in the subcritical equation (P-epsilon), the supercritical problem (P+epsilon) has no solutions which concentrate around a point of Omega as epsilon --> 0.