LAUSR

AbstractIn this paper, we study the following coupled Schrödinger system: {−Δu+u=u2∗−1+βu2∗2−1v2∗2+f(u),x∈RN,−Δv+v=v2∗−1+βu2∗2v2∗2−1+g(v),x∈RN,u,v>0,x∈RN,$$ \textstyle\begin{cases} -\Delta u+u=u^{2^{*}-1}+\beta u^{\frac{2^{*}}{2}-1}v^{\frac {2^{*}}{2}}+f(u), &x\in\mathbb{R}^{N}, \\ -\Delta v+v=v^{2^{*}-1}+\beta u^{\frac{2^{*}}{2}}v^{\frac {2^{*}}{2}-1}+g(v), &x\in\mathbb{R}^{N}, \\ u,v>0, &x\in\mathbb{R}^{N}, \end{cases} $$ where N≥5$N\geq5$ and 2∗=2NN−2$2^{*}=\frac{2N}{N-2}$. Note that the nonlinearity and the coupling terms are both of critical growth. Using the mountain pass theorem, Ekeland’s variational principle and the concentration-compactness principle, we show that this system has at least one positive least energy solution for each β∈(−1,0)∪(0,+∞)$\beta\in(-1,0)\cup (0,+\infty)$.