Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the ( p 1,…, p n ) -Laplacian

In this paper, we prove the existence of at least three weak solutions for the quasilinear elliptic systems { Δ p 1 u 1 + λ F u 1 ( x , u 1 , u 2 , … , u n ) = 0 in Ω , Δ p 2 u 2 + λ F u 2 ( x , u 1 , u 2 , … , u n ) = 0 in Ω , … Δ p n u n + λ F u n ( x , u 1 , u 2 , … , u n ) = 0 in Ω , u i = 0 for 1 ≤ i ≤ n on ∂ Ω . Our main tool is a recent three critical points theorem of Ricceri [B. Ricceri, On a three critical points theorem, Arch. Math. (Basel) 75 (2000) 220–226].