Positive solutions of slightly supercritical elliptic equations in symmetric domains

Abstract

This paper deals with existence and multiplicity of solutions for problem P(ε,Ω) below, which concentrate and blow-up at a finite number of points as ε→0. We give sufficient conditions on Ω which guarantee that the following property holds: there exists k―̄(Ω) such that, for each k⩾ k―̄(Ω), problem P(ε,Ω), for ε>0 small enough, has at least one solution blowing up as ε→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 Ω as k→∞. Moreover we present some examples which show that P(ε,Ω) may have k-spike solutions of this type also when Ω is a contractible domain, not necessarily close to domains with nontrivial topology and, for ε>0 small and k large enough, even when it is very close to star-shaped domains.