Concentration phenomena for solutions of superlinear elliptic problems

Abstract

In this paper we look for positive solutions of the problem −Δu+λu=up−1 in Ω, u=0 on ∂Ω, where Ω is a bounded domain in Rn, n⩾3, p>2 and λ is a positive parameter. We describe new concentration phenomena, which occur as λ→+∞, and exploit them to construct (for λ large enough) positive solutions that concentrate near spheres of codimension 2 as λ→+∞; these spheres approach the boundary of Ω as λ→+∞. Notice that the existence and multiplicity results we obtain hold also in contractible domains arbitrarily close to starshaped domains (no solution can exist if p⩾2nn−2 and Ω is starshaped, because of Pohožaev's identity). The method we use is completely variational and based on a blow up analysis in the equivariant setting. In order to avoid concentration phenomena near points and to overcome some difficulties related to the lack of compactness, we first modify the nonlinear term in a suitable region, then we solve the modified problem by minimizing the related energy functional on a suitable infinite dimensional manifold and, finally, we show that the solutions of the modified problem solve also our problem, for λ large enough, because they are localized in the prescribed region where the nonlinear term has not been modified.