Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
In this article, we study the following nonlinear Schrödinger system −Δu1+V1(x)u1=αu1u2+μu1,x∈R4,−Δu2+V2(x)u2=α2u12+βu22+μu2,x∈R4,\left\{\begin{array}{ll}-\Delta {u}_{1}+{V}_{1}\left(x){u}_{1}=\alpha {u}_{1}{u}_{2}+\mu {u}_{1},& x\in {{\mathbb{R}}}^{4},\\ -\Delta {u}_{2}+{V}_{2}\left(x){u}_{2}=\...
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| Format: | Article |
| Language: | English |
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De Gruyter
2024-01-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2023-0113 |
| _version_ | 1797340962952839168 |
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| author | Yang Jing |
| author_facet | Yang Jing |
| author_sort | Yang Jing |
| collection | DOAJ |
| description | In this article, we study the following nonlinear Schrödinger system −Δu1+V1(x)u1=αu1u2+μu1,x∈R4,−Δu2+V2(x)u2=α2u12+βu22+μu2,x∈R4,\left\{\begin{array}{ll}-\Delta {u}_{1}+{V}_{1}\left(x){u}_{1}=\alpha {u}_{1}{u}_{2}+\mu {u}_{1},& x\in {{\mathbb{R}}}^{4},\\ -\Delta {u}_{2}+{V}_{2}\left(x){u}_{2}=\frac{\alpha }{2}{u}_{1}^{2}+\beta {u}_{2}^{2}+\mu {u}_{2},& x\in {{\mathbb{R}}}^{4},\end{array}\right. with the constraint ∫R4(u12+u22)dx=1{\int }_{{{\mathbb{R}}}^{4}}\left({u}_{1}^{2}+{u}_{2}^{2}){\rm{d}}x=1, where α>0\alpha \gt 0 and α>β\alpha \gt \beta , μ∈R\mu \in {\mathbb{R}}, V1(x){V}_{1}\left(x), and V2(x){V}_{2}\left(x) are bounded functions. Under some mild assumptions on V1(x){V}_{1}\left(x) and V2(x){V}_{2}\left(x), we prove the existence of normalized peak solutions by using the finite dimensional reduction method, combined with the local Pohozaev identities. Because of the interspecies interaction between the components, we aim to obtain some new technical estimates. |
| first_indexed | 2024-03-08T10:11:04Z |
| format | Article |
| id | doaj.art-ace16386c26f4a4d9ca2d8d839520e46 |
| institution | Directory Open Access Journal |
| issn | 2191-950X |
| language | English |
| last_indexed | 2024-03-08T10:11:04Z |
| publishDate | 2024-01-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-ace16386c26f4a4d9ca2d8d839520e462024-01-29T08:47:33ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2024-01-0113140741210.1515/anona-2023-0113Existence of normalized peak solutions for a coupled nonlinear Schrödinger systemYang Jing0Department of Mathematics, Zhejiang University of Technology, 310014 Zhejiang, ChinaIn this article, we study the following nonlinear Schrödinger system −Δu1+V1(x)u1=αu1u2+μu1,x∈R4,−Δu2+V2(x)u2=α2u12+βu22+μu2,x∈R4,\left\{\begin{array}{ll}-\Delta {u}_{1}+{V}_{1}\left(x){u}_{1}=\alpha {u}_{1}{u}_{2}+\mu {u}_{1},& x\in {{\mathbb{R}}}^{4},\\ -\Delta {u}_{2}+{V}_{2}\left(x){u}_{2}=\frac{\alpha }{2}{u}_{1}^{2}+\beta {u}_{2}^{2}+\mu {u}_{2},& x\in {{\mathbb{R}}}^{4},\end{array}\right. with the constraint ∫R4(u12+u22)dx=1{\int }_{{{\mathbb{R}}}^{4}}\left({u}_{1}^{2}+{u}_{2}^{2}){\rm{d}}x=1, where α>0\alpha \gt 0 and α>β\alpha \gt \beta , μ∈R\mu \in {\mathbb{R}}, V1(x){V}_{1}\left(x), and V2(x){V}_{2}\left(x) are bounded functions. Under some mild assumptions on V1(x){V}_{1}\left(x) and V2(x){V}_{2}\left(x), we prove the existence of normalized peak solutions by using the finite dimensional reduction method, combined with the local Pohozaev identities. Because of the interspecies interaction between the components, we aim to obtain some new technical estimates.https://doi.org/10.1515/anona-2023-0113normalized peak solutionslocal pohozaev identitiesreduction method35a1035b9935j60 |
| spellingShingle | Yang Jing Existence of normalized peak solutions for a coupled nonlinear Schrödinger system Advances in Nonlinear Analysis normalized peak solutions local pohozaev identities reduction method 35a10 35b99 35j60 |
| title | Existence of normalized peak solutions for a coupled nonlinear Schrödinger system |
| title_full | Existence of normalized peak solutions for a coupled nonlinear Schrödinger system |
| title_fullStr | Existence of normalized peak solutions for a coupled nonlinear Schrödinger system |
| title_full_unstemmed | Existence of normalized peak solutions for a coupled nonlinear Schrödinger system |
| title_short | Existence of normalized peak solutions for a coupled nonlinear Schrödinger system |
| title_sort | existence of normalized peak solutions for a coupled nonlinear schrodinger system |
| topic | normalized peak solutions local pohozaev identities reduction method 35a10 35b99 35j60 |
| url | https://doi.org/10.1515/anona-2023-0113 |
| work_keys_str_mv | AT yangjing existenceofnormalizedpeaksolutionsforacouplednonlinearschrodingersystem |