Symmetries, Hopf fibrations and supercritical elliptic problems

2015/01/14 by Clapp, Mónica, Pistoia, Angela
#35J20 #35J25 #35J61 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1501.03349

Abstract

We consider the semilinear elliptic boundary value problem -Δu=\vert u\vert ^p-2u in Ω, u=0 on ∂Ω, in a bounded smooth domain Ω of ℝ^N for supercritical exponents p>(2N)/(N-2). Until recently, only few existence results were known. An approach which has been successfully applied to study this problem, consists in reducing it to a more general critical or subcritical problem, either by considering rotational symmetries, or by means of maps which preserve the Laplace operator, or by a combination of both. The aim of this paper is to illustrate this approach by presenting a selection of recent results where it is used to establish existence and multiplicity or to study the concentration behavior of solutions at supercritical exponents.

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