journal of differential equations 156, 407426 (1999) The Role Played by Space Dimension in Elliptic Critical Problems Enrico Jannelli Dipartimento di Matematica, Universita e Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy E-mail: jannellipascal.dm.uniba.it Received May 11, 1998; revised October 6, 1998 1. INTRODUCTION The celebrated paper [BN] deals with the problem &2u =u 2*&1 +*u in 0 u >0 in 0 (1.1) u =0 on 0, where 0 is a bounded smooth open subset of R N , N3, and 2*= (2NN&2) is the socalled critical exponent for Sobolev embedding. Pohozaev identity (see [P]) implies that (1.1) has no solution if 0 is strictly star shaped and * 0; on the other hand, the requirement u >0 in 0 implies * <* 1 , where * 1 is the first eigenvalue of &2 in 0. Hence, con- fined to 0<* <* 1 , the following results were proved in [BN]: Theorem A. If N4 then (1.1) has at least one solution u # H 1 0 ( 0) when 0<* <* 1 . Theorem B. If N=3, problem (1.1) has at least one solution u # H 1 0 ( 0) when * * <* <* 1 , where * * is a suitable positive number. Theorem C. If N=3 and 0 is a ball, then * * = 1 4 * 1 , and (1.1) has no solution for * * * . The preceding results show that the space dimension N plays a fundamental role when one seeks solutions of (1.1); in particular, the dimension N=3 is a special one, if compared with N4. According to the definition introduced by Pucci and Serrin (see [PS1; PS2]; see also [G]), we shall say that N=3 is a critical dimension for problem (1.1). More generally, we shall say that a dimension N is critical for a second order Article ID jdeq.1998.3589, available online at http:www.idealibrary.com on 407 0022-039699 30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.