On a K-component elliptic system with the Sobolev critical exponent in high dimensions: the repulsive case

Abstract

Study the following K-component elliptic system Here $$k\ge 2$$ 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 Brezis–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.