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The Editors-in-Chief have retracted this article [1] because it overlaps significantly with a number of previously published articles from different authors [2-4] and one article by different authors that was simultaneously under consideration with another journal [5]. The article also showed evidence of peer review manipulation. Additionally, the identity of the corresponding author could not be verified: Roskilde University have confirmed that Beatriz Ychussie has not been affiliated with their institution. The authors have not responded to any correspondence with regards to this retraction.

Retraction Note: Sharp geometrical properties of a-rarefied sets via fixed point index for the Schrödinger operator equations