We study the existence and multiplicity of solutions to the elliptic system displaylinesβhboxdiv(β£ablauβ£pβ2ablau)+m1β(x)β£uβ£pβ2u=lambdag(x,u)quadxinOmega,crβhboxdiv(β£ablavβ£pβ2ablav)+m2β(x)β£vβ£pβ2v=muh(x,v)quadxinOmega,crβ£ablauβ£pβ2fracpartialupartialn=fuβ(x,u,v),quadβ£ablavβ£pβ2fracpartialvpartialn=fvβ(x,u,v), where OmegasubsetmathbbRN is a bounded and smooth domain. Using fibering maps and extracting Palais-Smale sequences in the Nehari manifold, we prove the existence of at least two distinct nontrivial nonnegative solutions