COMMUNICATIONS ON Website: http://AIMsciences.org PURE AND APPLIED ANALYSIS Volume 2, Number 2, June 2003 pp. 171–186 NONHOMOGENEOUS POLYHARMONIC ELLIPTIC PROBLEMS AT CRITICAL GROWTH WITH SYMMETRIC DATA ∗ M´ onica Clapp Instituto de Matem´aticas Universidad Nacional Aut´onoma de M´ exico Circuito Exterior, Ciudad Universitaria 04510 M´ exico Marco Squassina Dipartimento di Matematica e Fisica Universit` a Cattolica del Sacro Cuore Via Musei 41, 25121, Brescia, Italy Abstract. We show the existence of multiple solutions of a perturbed poly- harmonic elliptic problem at critical growth with Dirichlet boundary conditions when the domain and the nonhomogenous term are invariant with respect to some group of symmetries. 1. Introduction and Main Result. Let K  1 and let Ω be a bounded smooth domain in R N with N> 2K. In this paper we consider the polyharmonic elliptic problem    (−∆) K u = |u| K ∗ −2 u + f in Ω, ( ∂ ∂ν ) j u    ∂Ω =0, j =0,...,K − 1, (P Ω,f ) where f ∈ H −K (Ω) and K ∗ = 2N N−2K denotes the critical exponent for the Sobolev embedding H K 0 (Ω)  → L K ∗ (Ω). If f = 0 this problem is invariant under dilations. Lack of compactness in elliptic problems which are invariant under dilations is known to produce quite interesting phenomena. It often gives rise to solutions of small perturbations of such problems. This behavior has been extensively studied for K = 1 (and 1 ∗ =2 ∗ ), we refer to [5], [27] and [30] for a detailed discussion. For K> 1 perturbations of problem (P Ω, 0 ) by adding a subcritical term have been considered by many authors; we refer to the work of Gazzola [14] and the references therein. For K = 2 perturbations of the domain giving rise to solutions were also recently considered by Gazzola, Grunau and one of the authors [15]. Adding a nonhomogeneous term produces a similar effect. For K = 1 it was shown by Tarantello [29] that, if f = 0 and ‖f ‖ H −1 is small enough, problem (P Ω,f ) has at least two nontrivial solutions. This result was extended to the case K = 2 by Deng and Wang [11]. One consequence of the main result in this paper is that this is true for every K  1. We shall show that the following holds. 1991 Mathematics Subject Classification. 31B30, 58E05. Key words and phrases. Polyharmonic problems, symmetric domains, multiplicity of solutions. * The first author was partially supported by CONACyT, Mexico under Research Project 28031- E. The second author was partially supported by MURST (40% – 1999) and by GNAFA. 171