Abstract In the present paper, we consider the following classes of elliptic systems with Sobolev critical growth: −Δu+λ1u=μ1u3+βuv2+2qpyu2qp−1v2inΩ,−Δv+λ2v=μ2v3+βu2v+2yu2qpvinΩ,u,v>0inΩ,u,v=0on∂Ω, $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta} u+\lambda_1u=\mu_1 u^3+\beta uv^2+\frac{2q}{p} y u^{\frac{2q}{p}-1}v^2\quad &\hbox{in}\;{\it\Omega}, \\ -{\it\Delta} v+\lambda_2v=\mu_2 v^3+\beta u^2v+2 y u^{\frac{2q}{p}}v\quad&\hbox{in}\;{\it\Omega}, \\ u,v \gt 0&\hbox{in}\;{\it\Omega}, \\ u,v=0&\hbox{on}\;\partial{\it\Omega}, \end{cases} \end{array}$$ where Ω ⊂ ℝ4 is a smooth bounded domain with smooth boundary ∂Ω; p, q are positive coprime integers with 1 < 2qp $\begin{array}{} \displaystyle \frac{2q}{p} \end{array}$ < 2; μi > 0 and λi ∈ ℝ are fixed constants, i = 1, 2; β > 0, y > 0 are two parameters. We prove a nonexistence result and the existence of the ground state solution to the above system under proper assumptions on the parameters. It seems that this system has not been explored directly before.

中文翻译：

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ℝ4 中的一类临界椭圆系统

摘要 在本文中，我们考虑以下具有 Sobolev 临界增长的椭圆系统： −Δu+λ1u=μ1u3+βuv2+2qpyu2qp−1v2inΩ,−Δv+λ2v=μ2v3+βu2v+2yu2qpvinΩ,u,v>0inΩ,u ,v=0on∂Ω, $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta} u+\lambda_1u=\mu_1 u^3+\beta uv^2+\frac{2q {p} yu^{\frac{2q}{p}-1}v^2\quad &\hbox{in}\;{\it\Omega}, \\ -{\it\Delta} v+\lambda_2v =\mu_2 v^3+\beta u^2v+2 yu^{\frac{2q}{p}}v\quad&\hbox{in}\;{\it\Omega}, \\ u,v \gt 0&\hbox{in}\;{\it\Omega}, \\ u,v=0&\hbox{on}\;\partial{\it\Omega}, \end{cases} \end{array}$$其中 Ω ⊂ ℝ4 是一个光滑有界域，其边界是光滑的∂Ω；p, q 是正互质整数，1 < 2qp $\begin{array}{} \displaystyle \frac{2q}{p} \end{array}$ < 2; μi > 0 和 λi ∈ ℝ 是固定常数，i = 1, 2；β > 0, y > 0 是两个参数。我们在参数的适当假设下证明了上述系统的不存在结果和基态解的存在。之前好像没有直接探索过这个系统。