Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.14115 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911377491230720 |
|---|---|
| author | Gao, Zijuan Guo, Qing Zhang, Chengxiang |
| author_facet | Gao, Zijuan Guo, Qing Zhang, Chengxiang |
| contents | 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 $N \geq 5$, $2^* = \frac{2N}{N-2}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14115 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Segregated Solutions to Critical Elliptic Systems in High Dimensions ($N \geq 5$) Gao, Zijuan Guo, Qing Zhang, Chengxiang Analysis of PDEs 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 $N \geq 5$, $2^* = \frac{2N}{N-2}$, 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. |
| title | Segregated Solutions to Critical Elliptic Systems in High Dimensions ($N \geq 5$) |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.14115 |