Solutions for a class of quasilinear Schrödinger equations with critical exponents term

In this paper, we study a class of quasilinear Schrödinger equation of the form $$-\varepsilon^2Δu+V(x)u-\varepsilon^2(Δ(|u|^{2α}))|u|^{2α-2}u &=&λ|u|^{q-2}u+|u|^{2^*(2α)-2}u,\quad\mbox{in}{\mathbb{R}}^N, $$ where $\varepsilon>0,λ>0,q\geq2,α>1/2$ are constants, $N\geq3$. By using change of variable and variational approach, the existence of positive solution which has a local maximum point and decays exponentially is obtained.

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