We exhibit new concentration phenomena for the equation -epsilon(2)Deltau + u = u(p) in a smooth bounded domain Omega subset of or equal to R-2 and with Neumann boundary conditions. The exponent p is greater than or equal to 2 and the parameter epsilon is converging to 0. For a suitable sequence epsilon(n) --> 0 we prove the existence of positive solutions u(n) concentrating at the whole boundary of Omega or at some component. (C) 2002 Wiley Periodicals, Inc.551215071568