Multiple solutions to the Bahri-Coron problem in domains with a shrinking hole of positive dimension

Author： Clapp Mónica  

Publisher： Taylor & Francis Ltd

ISSN： 1747-6933

Source： Complex Variables and Elliptic Equations, Vol.57, Iss.11, 2012-11, pp. : 1147-1162

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Abstract

Let Ω be a bounded smooth domain in R N , N ≥ 3, and let M be a compact smooth submanifold of Ω without boundary such that 1 ≤ dim M ≤ N − 2. We consider the problem where Ω ≔ {x ∈ Ω : dist(x, M) > }, is the critical Sobolev exponent, and > 0 is small enough. Under some additional assumptions we obtain the multiplicity of positive and sign changing solutions.

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