Segregated Solutions to Critical Elliptic Systems in High Dimensions ($N\ge 5$)

We study the existence of multiple segregated solutions to the critical coupled Schrödinger system $$\{\begin{array}{ll}-\mathrm{\Delta }{u}_1=K_1(\mathrm{\mid }y\mathrm{\mid })\mathrm{\mid }{u}_1{\mathrm{\mid }}^{2^{\ast }-2}{u}_1+\beta \mathrm{\mid }{u}_2{\mathrm{\mid }}^{\frac{2^{\ast }}{2}}\mathrm{\mid }{u}_1{\mathrm{\mid }}^{\frac{2^{\ast }}{2}-2}{u}_1,& y\in {\mathbb{R}}^N,\\ -\mathrm{\Delta }{u}_2=K_2(\mathrm{\mid }y\mathrm{\mid })\mathrm{\mid }{u}_2{\mathrm{\mid }}^{2^{\ast }-2}{u}_2+\beta \mathrm{\mid }{u}_1{\mathrm{\mid }}^{\frac{2^{\ast }}{2}}\mathrm{\mid }{u}_2{\mathrm{\mid }}^{\frac{2^{\ast }}{2}-2}{u}_2,& y\in {\mathbb{R}}^N,\\ u_1,u_2\ge 0,u_1,u_2\in C_0({\mathbb{R}}^N)\cap D^{1,2}({\mathbb{R}}^N),\end{array}$$ with $N\ge 5$, $2^{\ast }=\frac{2N}{N-2}$, radial potentials $K_1,K_2>0$,and repulsive coupling $\beta <0$.Under the assumption that $K_1$ and $K_2$ attain local maxima at distinct radii $r_0\mathrm{\ne }{\rho }_0$ with precise asymptotic expansions near these points, we prove the existence of infinitely many non-radial segregated solutions $(u_{1,k},u_{2,k})$ for all sufficiently large integers $k$. These solutions exhibit multiple bumps concentrating on two separate circles of radius $r_0$ and ${\rho }_0$ respectively. Moreover, each component develops a "dead core’’ near the concentration points of the other. The proof overcomes the sublinear and non-smooth nature of the coupling term ($2^{\ast }\mathrm{/}2-1<1$) by constructing a tailored complete metric space and combining a finite-dimensional reduction with a novel tail minimization argument.