A critical points approach for the existence of multiple solutions of a Dirichlet quasilinear system

In this paper, we establish the existence of at least three classical solutions for the Dirichlet quasilinear elliptic system{−(pi−1)|ui′(x)|pi−2ui″(x)=[λFui(x,u1,…,un)+μGui(x,u1,…,un)]hi(x,ui′),x∈(a,b),ui(a)=ui(b)=0,for i=1,…,n. Our main tool is a recent three critical points theorem of B. Ricceri [A three critical points theorem revisited, Nonlinear Anal. 70 (2009) 3084-3089].