Bull. London Math. Soc. 35 (2003) 513–521 C ❢ 2003 London Mathematical Society DOI: 10.1112/S0024609303001942 MULTI-BUBBLE SOLUTIONS FOR SLIGHTLY SUPER-CRITICAL ELLIPTIC PROBLEMS IN DOMAINS WITH SYMMETRIES MANUEL DEL PINO, PATRICIO FELMER and MONICA MUSSO Abstract The aim of this paper is to show the existence of solutions with an arbitrarily large number of bubbles for the slightly super-critical elliptic problem -∆u = u (N+2)/(N-2)+ε in Ω, subject to the conditions that u> 0 in Ω, and u = 0 on ∂Ω, where ε> 0 is a small parameter and Ω ⊂  N is a bounded domain with certain symmetries, for instance an annulus or a torus in  3 . 1. Introduction This paper deals with the construction of solutions to the elliptic problem    -∆u = u (N+2)/(N-2)+ε , in Ω, u> 0, in Ω, u =0, on ∂Ω, (1.1) where Ω is a smooth bounded domain in  N , N  3, and ε> 0 is a small parameter. It is well known that the transition in the power nonlinearity from below to above the critical exponent (N + 2)/(N - 2) implies a dramatic change in the structure of the solution set of this problem. For instance, from Pohozaev’s identity [14], it follows that no solution to (1.1) exists if Ω is star-shaped and ε  0, while for ε< 0 the problem is always solvable. The geometry and topology of the domain play a crucial role in the issues of the solvability and structure of the solution set when ε  0. This question has been addressed in numerous works. In [10], Kazdan and Warner showed the existence of a radially symmetric solution when the domain is an annulus. Coron [7] found that (1.1) is solvable when ε =0 in any domain exhibiting a small hole. Bahri and Coron [3] notably extended this result, proving that if ε = 0 and some homology group of Ω with coefficients in  2 is not trivial, then (1.1) has at least one solution, in particular in any three-dimensional domain that is not contractible to a point. Examples showing that this condition is actually not necessary for solvability were found by Dancer [8] and Ding [9]. On the other hand, Passaseo in [11] found examples of domains with non-trivial topology for which (1.1) is not solvable in cases where the power is sufficiently large, answering negatively a question posed by Brezis [5]. Nevertheless, the question of existence remained open for a power that is super- critical, but close to critical. Dancer conjectured that the solvability of (1.1) in a domain with nontrivial topology persists for small ε> 0. In [12] and [13] the authors showed that a solution to (1.1) exists for any small enough ε> 0 in Coron’s setting; that is, when Ω is a smooth domain with Received 23 November 2001; revised 13 May 2002. 2000 Mathematics Subject Classification 35J25 (primary); 35J20, 35J60 (secondary).