This paper is concerned with the concentration of positive ground states solutions for a modified Schrödinger equation $$\begin{aligned} -\varepsilon ^{2}\triangle u+V(x)u-\varepsilon ^{2}\triangle (u^{2})u=K(x)|u|^{p-2}u+|u|^{22^{*}-2}u,\quad \text{ in } \; \mathbb {R}^{N}, \end{aligned}$$ - ε 2 ▵ u + V ( x ) u - ε 2 ▵ ( u 2 ) u = K ( x ) | u | p - 2 u + | u | 22 ∗ - 2 u , in R N , where $$4<p<22^{*}, \varepsilon >0$$ 4 < p < 22 ∗ , ε > 0 is a parameter and $$2^{*}:=\frac{2N}{N-2}(N\ge 3)$$ 2 ∗ : = 2 N N - 2 ( N ≥ 3 ) is the critical Sobolev exponent. We prove the existence of a positive ground state solution $$v_{\varepsilon }$$ v ε and $$\varepsilon $$ ε sufficiently small under some suitable conditions on the nonnegative functions V(x) and K(x). Moreover, $$v_{\varepsilon }$$ v ε concentrates around a global minimum point of V as $$\varepsilon \rightarrow 0$$ ε → 0 . The proof of the main result is based on minimax theorems and concentration compact theory.