  In this paper, we consider the following two-component elliptic system with
critical growth \begin{equation*}
  \begin{cases}
  -\Delta u+(V_1(x)+\lambda)u=\mu_1u^{3}+\beta uv^{2}, \ \ x\in \mathbb{R}^4,
  -\Delta v+(V_2(x)+\lambda)v=\mu_2v^{3}+\beta vu^{2}, \ \ x\in \mathbb{R}^4 ,
  % u\geq 0, \ \ v\geq 0 \ \text{in} \ \R^4.
  \end{cases} \end{equation*} where $V_j(x) \in L^{2}(\mathbb{R}^4)$ are
nonnegative potentials and the nonlinear coefficients $\beta ,\mu_j$, $j=1,2$,
are positive. Here we also assume $\lambda>0$. By variational methods combined
with degree theory, we prove some results about the existence and multiplicity
of positive solutions under the hypothesis $\beta>\max\{\mu_1,\mu_2\}$. These
results generalize the results for semilinear Schr\"{o}dinger equation on half
space by Cerami and Passaseo (SIAM J. Math. Anal., 28, 867-885, (1997)) to the
above elliptic system, while extending the existence result from Liu and Liu
(Calc. Var. Partial Differential Equations, 59:145, (2020)).
