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

TL;DR

This paper proves the existence of infinitely many non-radial segregated solutions with multiple concentration points for a critical coupled Schrödinger system in high dimensions, using advanced variational methods.

Contribution

It introduces a novel approach to handle non-smooth, sublinear coupling terms and constructs solutions concentrating on two separate circles.

Findings

Existence of infinitely many solutions with multiple bumps

Solutions concentrate on two distinct circles

Components develop dead cores near each other's concentration points

Abstract

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.