Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the (p1,…,pn) -Laplacian

In this paper, we prove the existence of at least three weak solutions for the quasilinear elliptic systems {Δp1u1+λFu1(x,u1,u2,…,un)=0in Ω,Δp2u2+λFu2(x,u1,u2,…,un)=0in Ω,…Δpnun+λFun(x,u1,u2,…,un)=0in Ω,ui=0 for 1≤i≤non ∂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].