A nonlinear Lyapunov-Schmidt reduction and multiple solutions for some semilinear elliptic equation Pierpaolo ESPOSITO ∗ and Gianni MANCINI † e-mail: pesposit@axp.mat.uniroma2.it e-mail: mancini@mat.uniroma3.it Communicated by the second author in...................... Abstract We present a finite dimensional reduction for perturbed variational functionals and discuss some nonlinear elliptic PDE with Sobolev critical growth in bounded domains. 1 Introduction We will present in this talk a general variational principle for perturbative problems in presence of a manifold of ”quasi critical points” for the unperturbed energy functional. A model problem is the following: (P )      −△ u = u p + f (δ,x,u) in Ω ⊂ BbbR N ,N ≥ 3 u =0 on ∂ Ω u> 0 in Ω where p = N+2 N−2 is the limiting Sobolev exponent for the immersion of H 1 0 (Ω) in L q (Ω), q ≥ 1. Here f (δ,x,u) is a perturbation term, small if δ is small, satisfying the growth condition ∃ c>o : |f (δ,x,u)|≤ c(1 + |u| p ). * Universit` a di Roma Tor Vergata, Dipartimento di Matematica, via della Ricerca Scientifica, Roma 00133 Italy. † Universit` a degli Studi Roma Tre, Dipartimento di Matematica, Largo San Leonardo Murialdo 1, Roma 00146 Italy. The author’s research is supported by M.U.R.S.T. under the national project Variational methods and nonlinear differential equations