On least energy solutions for a nonlinear Schrödinger system with $K$-wise interaction
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918140308357120 |
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| author | Giaretto, Lorenzo Soave, Nicola |
| author_facet | Giaretto, Lorenzo Soave, Nicola |
| contents | In this paper we establish existence and properties of minimal energy solutions for the weakly coupled system $$ \begin{cases}
-Δu_i + λ_i u_i = μ_i|u_i|^{Kq-2}u_i + β|u_i|^{q-2}u_i\prod_{j\neq i}|u_j|^q & \text{in }\mathbb{R}^d, \qquad
u_i \in H^1(\mathbb{R}^d), \end{cases}\qquad i=1,\dots, K, $$ characterized by $K$-wise interaction (namely the interaction term involves the product of all the components). We consider both attractive ($β>0$) and repulsive cases ($β<0$), and we give sufficient conditions on $β$ in order to have least energy fully non-trivial solutions, if necessary under a radial constraint. We also study the asymptotic behavior of least energy fully non-trivial radial solutions in the limit of strong competition $β\to -\infty$, showing partial segregation phenomena which differ substantially from those arising in pairwise interaction models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03480 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On least energy solutions for a nonlinear Schrödinger system with $K$-wise interaction Giaretto, Lorenzo Soave, Nicola Analysis of PDEs 35R35, 35B25 (Primary) 35J61, 35J47 (Secondary) In this paper we establish existence and properties of minimal energy solutions for the weakly coupled system $$ \begin{cases} -Δu_i + λ_i u_i = μ_i|u_i|^{Kq-2}u_i + β|u_i|^{q-2}u_i\prod_{j\neq i}|u_j|^q & \text{in }\mathbb{R}^d, \qquad u_i \in H^1(\mathbb{R}^d), \end{cases}\qquad i=1,\dots, K, $$ characterized by $K$-wise interaction (namely the interaction term involves the product of all the components). We consider both attractive ($β>0$) and repulsive cases ($β<0$), and we give sufficient conditions on $β$ in order to have least energy fully non-trivial solutions, if necessary under a radial constraint. We also study the asymptotic behavior of least energy fully non-trivial radial solutions in the limit of strong competition $β\to -\infty$, showing partial segregation phenomena which differ substantially from those arising in pairwise interaction models. |
| title | On least energy solutions for a nonlinear Schrödinger system with $K$-wise interaction |
| topic | Analysis of PDEs 35R35, 35B25 (Primary) 35J61, 35J47 (Secondary) |
| url | https://arxiv.org/abs/2507.03480 |