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We consider the following quasilinear Schrodinger equations of the form $$\begin{aligned} \triangle u-\varepsilon V(x)u+u\triangle u^2+u^{p}=0,\ u>0\ \text{ in }\ \mathbb {R}^N\ \text{ and }\ \underset{|x|\rightarrow \infty }{\lim } u(x)=0, \end{aligned}$$ where $$N\ge 3,$$ $$p>\frac{N+2}{N-2},$$ $$\varepsilon >0$$ and V(x) is a positive function. By imposing appropriate conditions on V(x),  we prove that, for $$\varepsilon =1,$$ the existence of infinity many positive solutions with slow decaying $$O(|x|^{-\frac{2}{p-1}})$$ at infinity if $$p>\frac{N+2}{N-2}$$ and, for $$\varepsilon $$ sufficiently small, a positive solution with fast decaying $$O(|x|^{2-N})$$ if $$\frac{N+2}{N-2}<p<\frac{3N+2}{N-2}.$$ The proofs are based on perturbative approach. To this aim, we also analyze the structure of positive solutions for the zero mass problem.

Fast and slow decaying solutions for $$H^{1}$$-supercritical quasilinear Schrödinger equations