Proceedings of the Royal Society of Edinburgh, 131A, 1113{1132, 2001 A biharmonic problem with constraint involving critical Sobolev exponent M. Guedda, R. Hadiji and C. Picard Lamfa, CNRS UPRES-A 6119, Universit´ e de Picardie Jules Verne, Facult´ e de Math´ ematiques et d’Informatique, 33 rue Saint-Leu 80039 Amiens, France (MS received 30 September 1999; accepted 24 October 2000) We are concerned with the following minimization problems, inf « (¢u) 2 ; u 2 H (« ); « ju + ’j q c =1 ; where « » R N , N> 4, is a smooth bounded domain, qc =2N=(N ¡ 4), ’ 2 C (« ) \ L q c (« ) and H («)= H 2 («) \ H 1 0 (« ) or H 2 0 (« ). We show that, for ’ 6² 0, each in¯mum is achieved. Under suitable conditions on ’, we establish the following gap phenomenon, inf « j ¢uj 2 ; u 2 H 2 (« ) \ H 1 0 (« ); « ju + ’j q =1 < inf « (¢u) 2 ; u 2 H 2 0 (« ); « j u + ’j q =1 for q 6 qc . Moreover, we study the limit behaviour of the minimizers, as q goes to qc , in the case ’ 2 H («). 1. Introduction Let « be a smooth bounded domain of R N , N> 4. Let q c =2N=(N ¡ 4) be the limiting Sobolev exponent for the embedding H 2 (« ) » L q («). Let us consider the two minimization problems S («) d ef = inf »Z « (¢u) 2 ; u 2 H 2 0 («); Z « juj qc =1 ¼ (1.1) and S  («) d ef = inf »Z « (¢u) 2 ; u 2 H 2  («); Z « juj qc =1 ¼ ; (1.2) where H 2  («)= H 2 («) \ H 1 0 («) and ¢ denotes the Laplacian operator. It is well known [8,13,14] that the in-ma S (« ) and S  («) are never achieved, S («) and S  («) are independent of « and S (« )= S  («)= S ; the constant S is given by [12], S = º 2 N (N ¡ 4)(N 2 ¡ 4)  ¡ ( 1 2 N ) ¡ (N ) ´ 4=N : 1113 c ® 2001 The Royal Society of Edinburgh