Supercritical problems in domains with thin toroidal holes

TL;DR

This paper investigates the existence and multiplicity of sign-changing solutions to a supercritical Lane-Emden-Fowler equation in domains with thin toroidal holes, showing solutions increase as the holes shrink under symmetry conditions.

Contribution

It establishes the existence of multiple sign-changing solutions for a supercritical PDE in domains with thin toroidal holes, extending understanding of solution behavior in complex geometries.

Findings

Number of sign-changing solutions increases as the hole size decreases.

Solutions are obtained under symmetry assumptions.

The problem involves supercritical exponents related to the domain's topology.

Abstract

In this paper we study the Lane-Emden-Fowler equation $$(P)_\epsilon\ \{\Delta u+|u|^{q-2}u=0 \ \hbox{in}\ \mathcal D_\epsilon, u=0 \ \hbox{on}\ \partial\mathcal D_\epsilon.$$ Here $\mathcal D_\epsilon = \mathcal D \setminus \{x \in \mathcal D \ : \ \mathrm{dist}(x,\Gamma_\ell)\le \epsilon\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $\Gamma_\ell$ is an $\ell-$dimensional closed manifold such that $\Gamma_\ell \subset \mathcal D$ with $1\le \ell \le N-3$ and $q={2(N-\ell)\over N-\ell-2}$. We prove that, under some symmetry assumptions, the number of sign changing solutions to $(P)_\epsilon$ increases as $\epsilon$ goes to zero.