In this paper, we consider the following two-component elliptic system with critical growth \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+(V_1(x)+\lambda )u=\mu _1u^{3}+\beta uv^{2}, \ \ x\in {{\mathbb {R}}}^4, \\ -\Delta v+(V_2(x)+\lambda )v=\mu _2v^{3}+\beta vu^{2}, \ \ x\in {{\mathbb {R}}}^4, \\ \end{array}\right. } \end{aligned}\) 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 Schrödinger 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 Differ Equ 59:145, 2020).
