Nonlinear elliptic equations with large supercritical exponents

Abstract

The paper deals with the existence of positive solutions of the problem -Δ u=up in Ω, u=0 on ∂Ω, where Ω is a bounded domain of \(\mathcal{R}^n\), n≥ 3, and p>2. We describe new concentration phenomena, which arise as p→ +∞ and can be exploited in order to construct, for p large enough, positive solutions that concentrate, as p→ +∞, near submanifolds of codimension 2. In this paper we consider, in particular, domains with axial symmetry and obtain positive solutions concentrating near (n-2)-dimensional spheres, which approach the boundary of Ω as p→ +∞. The existence and multiplicity results we state allow us to find positive solutions, for large p, also in domains which can be contractible and even arbitrarily close to starshaped domains (while no solution can exist if Ω is starshaped and \(p\ge {2n\over n-2}\), as a consequence of the Pohožaev's identity).