Semi-classical states for a class of quasilinear Schrödinger equations with a parameter

In this paper, we study the following quasilinear Schrödinger equation with a parameter: $-{\epsilon }^2\mathrm{\Delta }u+V(x)u-{\epsilon }^2\kappa \beta \mathrm{\Delta }(|u{|}^{2\beta })|u{|}^{2\beta -2}u=|u{|}^{q-2}u+|u{|}^{(2\beta )2^{\ast }-2}u$ in ${\mathbb{R}}^N$, where $N⩾3$, $\beta >\frac{1}{2}$, $4\beta <q<(2\beta )2^{\ast }$, $\kappa$ is a constant, and $V:{\mathbb{R}}^N\to \mathbb{R}$ satisfies the classical global assumption. By using a change of variable, we obtain the existence, multiplicity, and concentration behavior. We will see that the existence of solutions depends heavily on the parameter $\beta$.