This work concerns with the existence and multiplicity of positive solutions for the following quasilinear Schrödinger equation
$ -\Delta u+V(x)u-\Delta(u^2)u = a(\epsilon x)g(u), \; \; \; \; x\in\mathbb R^N, $
where $ V(x)>0 $, $ u>0 $, $ a $ and $ g $ are continuous functions and $ a $ has $ m $ maximum points. With the change of variables we show that this equation has at least $ m $ nontrivial solutions by using variational methods, the Ekeland's variational principle, and some properties of the Nehari manifold. Some recent results are improved.
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