In this paper, we concern the modified Schr\“{o}dinger equations $$\ -{\varepsilon}^{2}\Delta u+V(x)u-\varepsilon^{2}u\Delta u^2=|u|^{22^*-2}u+g(u), \ x \ \in \mathbb{R}^N.$$ First, a existence result of ground state positive solutions is given. Next, we research multiplicity and concentration of positive solutions. Where $N\geq 2$, $\varepsilon$ is positive parameters and $2^*=\frac{2N}{N-2}$ is the critical exponent, $V \in C(\mathbb{R}^N, \mathbb{R^{+}})$, $g \in C(\mathbb{R}, \mathbb{R})$. Our results improve corresponding results in \cite{HQZ} (X. He, A. Qian, W. Zou, Existence and concentration of positive solutions for quasilinear Schr\”{o}dinger equations with critical growth, Nonlinearity, 26(2013), 3137-3168).

In this paper, we study the following quasilinear Choquard equations of the form $$\ -\Delta u+V(x)u-\Delta (|u|^{2\alpha})|u|^{2\alpha-2}u=(|x|^{-\mu}\ast G(u))g(u), \ x \ \in R^N,$$ where $1\geq\alpha>\frac{1}{2}$, $V \in C(\mathbb{R}^N, \mathbb{R})$, $g \in C(\mathbb{R}^N, \ \mathbb{R})$. Distinguished from two situations $\lim\limits_{|x|\rightarrow\infty}V(x)=+\infty$ or $\lim\limits_{|x|\rightarrow\infty}V(x)<+\infty$, we research the existence of nontrivial solutions and a sequence of high energy solutions.