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

TL;DR

This work establishes the existence of infinitely many nonradial segregated solutions for a high-dimensional critical coupled Schrödinger system with repulsive coupling $\beta<0$ and radial potentials $K_1,K_2$, in dimensions $N\ge 5$ where the critical exponent is $2^* = \frac{2N}{N-2}$. By concentrating on two circles of radii $r_0$ and $\rho_0$ where $K_1$ and $K_2$ attain local maxima, the authors construct a tailored complete metric space and combine a finite-dimensional reduction with a tail-minimization argument to handle the sublinear, non-smooth coupling. They develop precise outer-region minimization, decay estimates, and dead-core analysis, then implement a fixed-point reduction and a Miranda-type zero search to obtain parameter choices yielding exact solutions. The resulting solutions exhibit multiple bumps concentrating on the two circles, with each component vanishing near the other's concentration points (dead cores), and the method yields infinitely many such segregated states as the mode index grows. This advances the understanding of high-dimensional critical systems with sublinear intercomponent coupling and demonstrates a robust reduction framework applicable to non-radial, multi-bump configurations.

Abstract

We study the existence of multiple segregated solutions to the critical coupled Schrödinger system $-Δu_1 = K_1(| y|) \lvert u_{1}\rvert^{2^{*}-2}u_{1}+β\lvert u_{2}\rvert^{\frac{2^{*}}{2}}\lvert u_{1}\rvert^{\frac{2^{*}}{2}-2}u_{1}, -Δu_{2} = K_2(\lvert y\rvert) \lvert u_{2}\rvert^{2^{*}-2}u_{2}+β\lvert u_{1}\rvert^{\frac{2^{*}}{2}}\lvert u_{2}\rvert^{\frac{2^{*}}{2}-2}u_{2}, y\in\mathbb R^N, u_{1},u_{2}\geq0,u_{1},u_{2}\in C_{0}(\mathbb R^{N})\cap D^{1,2}(\mathbb R^N),$ with $N \geq 5$, $2^* = \frac{2N}{N-2}$, radial potentials $K_1, K_2 > 0$,and repulsive coupling $β< 0$.Under the assumption that $K_1$ and $K_2$ attain local maxima at distinct radii $r_0 \ne ρ_0$ 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 $ρ_0$ 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.

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• Remark 2.4
• proof : Proof of Lemma \ref{['L2.2']}
• Lemma 2.5