We study the existence of multiple segregated solutions to the critical coupled Schr\"odinger system \[ \begin{ cases} -\Delta u_{ 1} = K_1(| y|) | u_{ 1}|^{ 2^*-2}u_{ 1}+\beta | u_{ 2}|^{ \frac{ 2^{ *}}{ 2}}| u_{ 1}|^{ \frac{ 2^{ *}}{ 2}-2}u_{ 1}, & y\in \mathbb R^N,\\ -\Delta u_{ 2} = K_2(| y|) | u_{ 2}|^{ 2^*-2}u_{ 2}+\beta | u_{ 1}|^{ \frac{ 2^{ *}}{ 2}}| u_{ 2}|^{ \frac{ 2^{ *}}{ 2}-2}u_{ 2}, & y\in\mathbb R^N,\\ u_{ 1},u_{ 2}\geq0, u_{ 1},u_{ 2}\in C_0(\mathbb R^{ N})\cap D^{ 1,2}(\mathbb R^N), \end{ cases} \] with $N \geq 5$, $2^* = \frac{ 2N}{ N-2}$, 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 $r_0 \ne \rho_0$ 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 ($2^*/2 -1 < 1$) by constructing a tailored complete metric space and combining a finite-dimensional reduction with a novel tail minimization argument.
Categories: math.AP