Solutions for a class of quasilinear Schr\"{o}dinger equations with critical exponents term

Abstract

In this paper, we study a class of quasilinear Schr\"{o}dinger equation of the form $$-\varepsilon^2\Delta u+V(x)u-\varepsilon^2(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u &=&\lambda|u|^{q-2}u+|u|^{2^*(2\alpha)-2}u,\quad\mbox{in}{\mathbb{R}}^N, $$ where $\varepsilon>0,\lambda>0,q\geq2,\alpha>1/2$ 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.