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We consider the following quasi-linear Schrödinger equation−∑ i, j= 1 N D j (a i j (u) D i u)+ 1 2∑ i, j= 1 N D s a i j (u) D i u D j u+ V (x) u= f (u), x∈ R N, N≥ 3, which includes the modified nonlinear Schrödinger equations. Combining a p-Laplacian perturbation argument, we obtain the existence of positive solutions to the problem above and multiple solutions with additional symmetry conditions on a i j and f. A new perturbation approach is used to treat the sub-cubic nonlinearity. In particular, for f (u)=| u| p− 2 u, the case p∈(2, 4) is also considered.