Positive solutions for a quasilinear Schr¨odinger equation with critical growth ∗ Giovany M. Figueiredo Universidade Federal do Par´ a, Departamento de Matem´ atica 66075-100, Bel´ em-PA, Brazil giovany@ufpa.br Marcelo F. Furtado Universidade de Bras´ ılia, Departamento de Matem´atica 70910-900, Bras´ ılia-DF, Brazil mfurtado@unb.br In memorian of Prof. Jack Hale Abstract We consider the quasilinear problem −ε p div(|∇u| p−2 ∇u)+ V (z )u p−1 = f (u)+ u p * −1 , u ∈ W 1,p (R N ), where ε> 0 is a small parameter, 1 <p<N , p ∗ = Np/(N − p), V is a positive potential and f is a superlinear function. Under a local condition for V we relate the number of positive solutions with the topology of the set where V attains its minimum. In the proof we apply Ljusternik-Schnirelmann theory. 2000 Mathematics Subject Classification : 35J50, 35B33, 58E05. Key words : Quasilinear Schr¨odinger equation; Ljusternik-Schnirelmann theory; Posi- tive solutions; Critical problems. 1 Introduction The main purpose of this paper is to establish a multiplicity result for the following quasi- linear critical problem (P ε )  −ε p Δ p u + V (z )u p−1 = f (u)+ u p * −1 in R N , u ∈ C 1,α loc (R N ) ∩ W 1,p (R N ),u(z ) > 0 for all z ∈ R N , * The authors were partially supported by CNPq/Brazil 1