Symmetries, Hopf fibrations and supercritical elliptic problems

TL;DR

This paper explores advanced mathematical techniques involving symmetries and fibrations to establish existence, multiplicity, and concentration properties of solutions for supercritical elliptic boundary value problems.

Contribution

It demonstrates how symmetry and geometric methods can be used to analyze supercritical elliptic problems, extending the understanding of solution existence and behavior.

Findings

Existence of solutions for supercritical exponents.

Multiplicity of solutions under symmetry constraints.

Concentration phenomena at high exponents.

Abstract

We consider the semilinear elliptic boundary value problem \[ -\Delta u=\left\vert u\right\vert ^{p-2}u\text{ in }\Omega,\text{\quad }u=0\text{ on }\partial\Omega, \] in a bounded smooth domain $\Omega$ of $\mathbb{R}^{N}$ for supercritical exponents $p>\frac{2N}{N-2}.$ 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.