On a <Emphasis Type="Italic">K</Emphasis>-component elliptic system with the Sobolev critical exponent in high dimensions: the repulsive case

Study the following K-component elliptic system

Open image in new window\(k\ge 2\) is a integer and \(\Omega \subset \mathbb {R}^N(N\ge 4)\) is a bounded domain with smooth boundary \(\partial \Omega \), \(a_i,\lambda _i>0\), \(b_i\ge 0\) for all \(i=1,2,\ldots ,k\) and \(\beta <0\), \(2^*=\frac{2N}{N-2}\) is the Sobolev critical exponent. By the variational method, we obtain a nontrivial solution of this system. The concentration behavior of this nontrivial solution as \(\overrightarrow{\mathbf {b}}\rightarrow \overrightarrow{\mathbf {0}}\) and \(\beta \rightarrow -\infty \) are both studied and the phase separation is exhibited for \(N\ge 6\), where \(\overrightarrow{\mathbf {b}}=(b_1,b_2,\ldots ,b_k)\) is a vector. Our results extend and generalize the results in Chen and Zou  (Arch Ration Mech Anal 205:515–551, 2012; Calc Var Partial Differ Equ 52:423–467, 2015). Moreover, by studying the phase separation, we also prove some existence and multiplicity results of the sign-changing solutions to the following Brezís–Nirenberg problem of the Kirchhoff type

$$\begin{aligned} \left\{ \begin{array}{ll} -\bigg (a+b\int _{\Omega }|\nabla u|^2dx\bigg )\Delta u = \lambda u +|u|^{2^*-2}u, &{}\quad \text {in }\Omega , \\ u =0,&{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$

where \(N\ge 6\), \(a,\lambda >0\) and \(b\ge 0\). These results can be seen as an extension of the results in Cerami et al. (J Funct Anal 69:289–306, 1986). The concentration behaviors of the sign-changing solutions to the above equation as \(b\rightarrow 0^+\) are also obtained.