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

Abstract

We study the existence of multiple segregated solutions to the critical coupled Schrödinger system \[ \begin{cases} -Δu_{1} = K_1(| y|) | u_{1}|^{2^*-2}u_{1}+β| u_{2}|^{\frac{2^{*}}{2}}| u_{1}|^{\frac{2^{*}}{2}-2}u_{1}, & y\in \mathbb R^N,\\ -Δu_{2} = K_2(| y|) | u_{2}|^{2^*-2}u_{2}+β| u_{1}|^{\frac{2^{*}}{2}}| u_{2}|^{\frac{2^{*}}{2}-2}u_{2}, & y\in\mathbb R^N,\\ u_{1},u_{2}\geq0, u_{1},u_{2}\in C_0(\mathbb R^{N})\cap D^{1,2}(\mathbb R^N), \end{cases} \] with $N\ge 5$, $2^{\ast }=\frac{2N}{N-2}$, 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 $r_0\ne {\rho }_0$ 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 ($2^{\ast }/2-1<1$) by constructing a tailored complete metric space and combining a finite-dimensional reduction with a novel tail minimization argument.