Positive solutions of slightly supercritical elliptic equations in symmetric domains

This paper deals with existence and multiplicity of solutions for problem P(epsilon, Omega) below, which concentrate and blow-up at a finite number of points as epsilon --> 0. We give sufficient conditions on Omega which guarantee that the following property holds: there exists (k) over bar(Omega) such that, for each k greater than or equal to (k) over bar(Omega), problem P(epsilon, Omega), for epsilon > 0 small enough, has at least one solution blowing up as epsilon --> 0 at exactly k points. Exploiting the properties of the Green and Robin functions, we also prove that the blow up points approach the boundary of Omega as k --> infinity. Moreover we present some examples which show that P(epsilon, Omega) may have k-spike solutions of this type also when Omega is a contractible domain, not necessarily close to domains with nontrivial topology and, for epsilon > 0 small and k large enough, even when it is very close to star-shaped domains. (C) 2004 Elsevier SAS. All rights reserved.