Journal of Differential Equations 182, 511–540 (2002) doi:10.1006/jdeq.2001.4098 Multi-Peak Solutions for Super-Critical Elliptic Problems in Domains with Small Holes Manuel del Pino and Patricio Felmer 1 Departamento de Ingenier ! ıa Matem ! atica, and Centro de Modelamiento Matem ! atico, UMR2071 CNRS-UChile Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile E-mail: delpino@dim.uchile.cl; pfelmer@dim.uchile.cl and Monica Musso 2 Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Turin, Italy E-mail: musso@calvino.polito.it Received December 21, 2000 This paper deals with the slightly super-critical elliptic problem Du ¼ u ðNþ2Þ ðN2Þ þe in O u > 0 in O u ¼ 0 on @O; 8 > > < > > : ð1Þ where e > 0 is a small parameter and O  R N is a bounded domain with smooth boundary. Assuming that the domain exhibits k sufficiently small holes, multiple solutions are constructed by gluing double-spike patterns located near each of the holes. # 2002 Elsevier Science (USA) Key Words: supercritical exponent; solution with multiple double-spikes. 1. INTRODUCTION This paper deals with the construction of solutions of the problem Du ¼ u ðN þ2Þ ðN 2Þ þe in O; u > 0 in O; u ¼ 0 on @O; ð1Þ 1 Supported by Fondecyt Lineas Complementarias Grant 8000010 and FONDAP Matem! aticas Aplicadas, Chile. 2 Partially supported by INDAM, Italy. 511 0022-0396/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.