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arXiv · 1501.03349

Symmetries, Hopf fibrations and supercritical elliptic problems

Mónica Clapp·Angela Pistoia
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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.

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Mónica Clapp, Angela Pistoia. 2015-01-14. Symmetries, Hopf fibrations and supercritical elliptic problems. https://arxiv.org/abs/1501.03349

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