Concentration of low energy extremals: Identification of concentration points

We study the variational problem where , is a bounded domain, , F satisfies $0leq F|t|leq alpha |t|^{2^*}$ and is upper semicontinuous. We show that to second order in the value only depends on two ingredients. The geometry of enters through the Robin function (the regular part of the Green's function) and F enters through a quantity which is computed from (radial) maximizers of the problem in . The asymptotic expansion becomes Using this we deduce that a subsequence of (almost) maximizers of must concentrate at a harmonic center of : i.e., , where is a minimum point of . Received: 24 January 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001