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 { − ( p i − 1 ) | u i ′ ( x ) | p i − 2 u i ″ ( x ) = [ λ F u i ( x , u 1 , … , u n ) + μ G u i ( x , u 1 , … , u n ) ] h i ( x , u i ′ ) , x ∈ ( a , b ) , u i ( a ) = u i ( 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].