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Supercritical problems in domains with thin toroidal holes

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  • In this paper we study the Lane-Emden-Fowler equation $$ (P)_ \epsilon \quad \left\{ \begin{aligned} &\Delta u+|u|^{q-2}u=0\ &\hbox{in}\ \mathcal D_ \epsilon,\\ & u=0\ &\hbox{on}\ \partial\mathcal D_ \epsilon.\\ \end{aligned}\right. $$ Here $\mathcal D_ \epsilon=\mathcal D\setminus \left\{x\in \mathcal D\ :\ \mathrm{dist}(x,\Gamma_l)\le \epsilon \right\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $\Gamma_l$ is an $l-$dimensional closed manifold such that $\Gamma_l\subset\mathcal D$ with $1\le l\le N-3$ and $q={2(N-l)\over N-l-2} .$ We prove that, under some symmetry assumptions, the number of sign changing solutions to $ (P)_ \epsilon$ increases as $\epsilon$ goes to zero.
    Mathematics Subject Classification: Primary: 35J60; Secondary: 35J20.


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