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# Multiple solutions to the Bahri-Coron problem in a bounded domain without a thin neighborhood of a manifold

Ponente: Juan Carlos Fernández
Institución: IM-UNAM
Cuándo 20/11/2014 de 12:00 a 13:00 Salón de seminarios 1 vCal iCal

We show that the critical problem
$-\Delta u=|u|^{\frac{{4}}{{N-2}}}u\ \text{in }\Omega,\quad\ u=0\ \text{on }\partial\Omega,$
has at least
$\max\{\text{cat}(\Theta,\Theta\smallsetminus B_{r}M),\text{cupl}(\Theta,\Theta\smallsetminus B_{r}M)+1\}\geq2$
pairs of nontrivial solutions in every domain $$\Omega$$ obtained by deleting from a given bounded
smooth domain $$\Theta\subset\mathbb{R}^{N}$$ a thin enough tubular neighborhood $$B_{r}M$$ of a
closed smooth submanifold $$M$$ of $$\Theta$$ of dimension $$\leq N-2,$$ where "cat" is the Lusternik-
Schnirelmann category and "cupl" is the cup-length of the pair.
This is joint work with Mónica Clapp.