Positive solutions for coupled Schrödinger system with critical exponent in R N $\mathbb{R}^{N}$ ( N ≥ 5 $N≥5$ ) - Yan-Fang Peng; Hong-Yu Ye (Boundary Value Problems, 2017)
Abstract In this paper, we study the following coupled Schrödinger system: { − Δ u + u = u 2 ∗ − 1 + β u 2 ∗ 2 − 1 v 2 ∗ 2 + f ( u ) , x ∈ R N , − Δ v + v = v 2 ∗ − 1 + β u 2 ∗ 2 v 2 ∗ 2 − 1 + g ( v ) , x ∈ R N , u , v > 0 , x ∈ R N , $$ \textstyle\begin{cases} -\Delta u+u=u^{2^{*}-1}+\beta u^{\frac{2^{*}}{2}-1}v^{\frac {2^{*}}{2}}+f(u), &x\in\mathbb{R}^{N}, \\ -\Delta v+v=v^{2^{*}-1}+\beta u^{\frac{2^{*}}{2}}v^{\frac {2^{*}}{2}-1}+g(v), &x\in\mathbb{R}^{N}, \\ u,v>0, &x\in\mathbb{R}^{N}, \end{cases} $$ where N ≥ 5 $N\geq5$ and 2 ∗ = 2 N N − 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 , + ∞ ) $\beta\in(-1,0)\cup (0,+\infty)$ .