Concentration phenomena for the volume functional in unbounded domains: identification of concentration points

We study the variational problem [GRAPHICS] in possibly unbounded domains Qsubset ofR(n), where n greater than or equal to 3, 2* = (2n)/(n-2) and F satisfies 0less than or equal toF(t)less than or equal toalphat(2)* and is upper semicontinuous. Extending earlier results for bounded domains, we show that (almost) maximizers of S-epsilon(F) (Omega) concentrate at a harmonic center, i.e. a minimum point of the Robin function tau(Omega) (the regular part of the Green function restricted to the diagonal). Moreover, we obtain the asymptotic expansion [GRAPHICS] where S-F and winfinity depend only on F but not on Omega and can be computed from radial maximizers of the corresponding problem in R-n. The crucial point is to find a suitable definition Of tau(Omega)(infinity). Interestingly the correct definition may be different from the lower semicontinuous extension of tau(Omega)(Omega{infinity}) to infinity, at least for n greater than or equal to 5. (C) 2003 Published by Elsevier Science (USA).