Ground States of a 𝐾-Component Critical System with Linear and Nonlinear Couplings: The Attractive Case

Abstract

Abstract Consider the system { - Δ ⁢ u i + μ i ⁢ u i = ν i ⁢ u i 2 * - 1 + β ⁢ ∑ j = 1 , j ≠ i k u j 2 * 2 ⁢ u i 2 * 2 - 1 + λ ⁢ ∑ j = 1 , j ≠ i k u j in ⁢ Ω , u i > 0 in ⁢ Ω , u i = 0 on ⁢ ∂ ⁡ Ω , i = 1 , 2 , … , k , \left\{\begin{aligned} \displaystyle-\Delta u_{i}+\mu_{i}u_{i}&\displaystyle=% \nu_{i}u_{i}^{2^{*}-1}+\beta\mathop{\sum_{j=1,j\neq i}^{k}}u_{j}^{\frac{2^{*}}% {2}}u_{i}^{\frac{2^{*}}{2}-1}+\lambda\mathop{\sum_{j=1,j\neq i}^{k}}u_{j}&&% \displaystyle\phantom{}\text{in}\ \Omega,\\ \displaystyle u_{i}&\displaystyle>0&&\displaystyle\phantom{}\text{in}\ \Omega,% \\ \displaystyle u_{i}&\displaystyle=0&&\displaystyle\phantom{}\text{on}\ % \partial\Omega,\quad i=1,2,\ldots,k,\end{aligned}\right. where k ≥ 2 {k\geq 2} , Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 3 {N\geq 3} ) is a bounded domain, 2 * = 2 ⁢ N N - 2 {2^{*}=\frac{2N}{N-2}} , μ i ∈ ℝ {\mu_{i}\in\mathbb{R}} and ν i > 0 {\nu_{i}>0} are constants, and β , λ > 0 {\beta,\lambda>0} are parameters. By showing a unique result of the limit system, we prove existence and nonexistence results of ground states to this system by variational methods, which generalize the results in [7, 18]. Concentration behaviors of ground states for β , λ {\beta,\lambda} are also established.