Multipeak solutions to the Bahri–Coron problem in domains with a shrinking hole

We construct positive and sign changing multipeak solutions to the pure critical exponent problem in a bounded domain with a shrinking hole, having a peak which concentrates at some point inside the shrinking hole (i.e. outside the domain) and one or more peaks which concentrate at interior points of the domain. These are, to our knowledge, the first multipeak solutions in a domain with a single small hole.