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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.13172 |
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| _version_ | 1866912489257566208 |
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| author | Jianjun, Zhang Xuexiu, Zhong Jinfang, Zhou |
| author_facet | Jianjun, Zhang Xuexiu, Zhong Jinfang, Zhou |
| contents | We are concerned with the following system of two coupled time-independent Gross-Pitaevskii equations $$ \begin{cases} -Δu+λ_1 u=μ_1|u|^{p-2}u+να|u|^{α-2}|v|^βu ~\hbox{in}~ \R^N,\\ -Δv+λ_2 v=μ_2|v|^{q-2}v+νβ|u|^α|v|^{β-2}v ~\hbox{in}~ \R^N, \end{cases} $$ which arises in two-components Bose-Einstein condensates and involve attractive Sobolev subcritical or critical interactions, i. e., $ν>0$ and $α+β\leq 2^*$. This system is employed by seeking critical points of the associated variational functional with the constrained mass below $$\int_{\mathbb{R}^N}|u|^2 {\rm d}x=a, \quad \int_{\mathbb{R}^N}|v|^2 {\rm d}x=b.$$ In the mass mixed case, i. e., $2<p<2+\frac{4}{N}<q<2^*$, for some suitable $a,b,ν$ and $β$, the system above admits two positive solutions. In particular, in the case $α+β<2^*$, using variational methods on the $L^2$-ball, two positive solutions are obtained, one of which is a local minimizer and the second one is a mountain pass solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13172 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple normalized solutions for two coupled Gross-Pitaevskii equations with attractive interactions and mass constriants Jianjun, Zhang Xuexiu, Zhong Jinfang, Zhou Analysis of PDEs We are concerned with the following system of two coupled time-independent Gross-Pitaevskii equations $$ \begin{cases} -Δu+λ_1 u=μ_1|u|^{p-2}u+να|u|^{α-2}|v|^βu ~\hbox{in}~ \R^N,\\ -Δv+λ_2 v=μ_2|v|^{q-2}v+νβ|u|^α|v|^{β-2}v ~\hbox{in}~ \R^N, \end{cases} $$ which arises in two-components Bose-Einstein condensates and involve attractive Sobolev subcritical or critical interactions, i. e., $ν>0$ and $α+β\leq 2^*$. This system is employed by seeking critical points of the associated variational functional with the constrained mass below $$\int_{\mathbb{R}^N}|u|^2 {\rm d}x=a, \quad \int_{\mathbb{R}^N}|v|^2 {\rm d}x=b.$$ In the mass mixed case, i. e., $2<p<2+\frac{4}{N}<q<2^*$, for some suitable $a,b,ν$ and $β$, the system above admits two positive solutions. In particular, in the case $α+β<2^*$, using variational methods on the $L^2$-ball, two positive solutions are obtained, one of which is a local minimizer and the second one is a mountain pass solution. |
| title | Multiple normalized solutions for two coupled Gross-Pitaevskii equations with attractive interactions and mass constriants |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.13172 |