Multi-bubble solutions for a slightly supercritical elliptic problem in a domain with a small hole

We are concerned with the existence and the asymptotic analysis when the parameter εtends to 0of solutions with multiple concentration for the following almost critical problem: −Δu = u^{(N+2)/(N−2)}+ε in Ω, u>0 inΩ, u= 0 on ∂Ω, where Ωis a bounded domain in RNwith a smooth boundary and N≥3. Weare interested in concentration phenomena from the supercritical side ε →0+. In particular we prove that, if Ωhas a small and not necessarily symmetric hole, then for any fixed odd integer k≥3 there exists a family of solutions which develops a multiple bubble-shape as ε →0+, blowing up at kdifferent points in Ω. This extends the previous result by Del Pino, Felmer and Musso [13], where solutions with a two-bubbleprofile are constructed.