Standing waves for two-component elliptic system with critical growth in R4: the attractive case

In this paper, we consider the following two-component elliptic system with critical growth (Formula presented.) where Vj(x)∈L2(R4) are nonnegative potentials and the nonlinear coefficients β,μj, j=1,2, are positive. Here we also assume λ>0. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of positive solutions under the hypothesis β>max{μ1,μ2}. 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).