We consider a class of p-Laplacian quasilinear Schrödinger equations { − ∆ p u − p 2 p − 1 u ∆ p ( u 2 )= λ u − γ + u q in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N is a bounded domain with regular boundary, 1 ∞, 0 1, 2 p − 1 < q ≤ 2 · p ∗ − 1 for p≤ N, 2 p−1 ∞ for p>N, where p ∗ = Np N − p if 1 , p ∗ ∈ ( p , ∞ ) is arbitrarily large if p= N, p ∗ = ∞ if p>N. We establish global existence and multiplicity results of positive solutions via a new strong comparison principle and a regularity result for weak solutions.