Multiplicity Results for a Quasilinear Elliptic System via Morse Theory SILVIA CINGOLANI Dipartimento di Matematica, Politecnico di Bari Via Amendola 126/B, 70126 Bari, Italy cingolan@poliba.it MONICA LAZZO Dipartimento di Matematica, Universit`a degli Studi di Bari Via Orabona 4, 70125 Bari, Italy lazzo@dm.uniba.it GIUSEPPINA VANNELLA Dipartimento di Matematica, Politecnico di Bari Via Amendola 126/B, 70126 Bari, Italy vannella@poliba.it Abstract In this work we prove some multiplicity results for solutions of a system of elliptic quasilinear equations, involving the p-Laplace operator (p> 2). The proofs are based on variational and topological arguments and make use of new perturbation results in Morse theory for the Banach space W 1,p 0 . MSC 2000: 58E05, 35B20, 35J50 1 Introduction Let us consider a system of linear Schr¨ odinger equations i ∂ Ψ ∂t = −  2 2m ΔΨ, (1) where  is the Planck constant, Ψ : Ω → R n+1 , Ω is a domain of R n , and n ≥ 3. It is known that the linear Schr¨ odinger equation is dispersive in nature, so that a wave packet disperses in a short time. However, if a nonlinear term is added in (1), localized finite energy wave packets (the so-called solitons) may exist. 1