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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.13163 |
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| _version_ | 1866916849133813760 |
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| author | Jianjun, Zhang Xuexiu, Zhong Jinfang, Zhou |
| author_facet | Jianjun, Zhang Xuexiu, Zhong Jinfang, Zhou |
| contents | We are concerned with qualitative properties of positive solutions to the following coupled Sobolev critical Schrödinger equations $$ \begin{cases} -Δu+λ_1 u=μ_1|u|^{2^*-2}u+να|u|^{α-2}|v|^βu ~\hbox{in}~ \R^N,\\ -Δv+λ_2 v=μ_2|v|^{2^*-2}v+νβ|u|^α|v|^{β-2}v ~\hbox{in}~ \R^N \end{cases} $$ subject to the mass constraints $\int_{\mathbb{R}^N}|u|^2 \ud x=a^2$ and $\int_{\mathbb{R}^N}|v|^2 \ud x=b^2$, where, $a>0,\,b>0,\,N=3,4$ and $2^*:=\frac{2N}{N-2}$ is the Sobolev critical exponent. The main purpose of this paper is focused on the mass mixed case, i. e., $ α>1,β>1,α+β<2+\frac{4}{N}$. For some suitable small $ν>0$, we show that the above system admits two positive solutions, one of which is a local minimizer, and another one is a mountain pass solution. Moreover, as $ν\to0^+$, asymptotic behaviors of solutions are also considered. Our result gives an affirmative answer to a Soave's type open problem raised by Bartsch {\it et al.} (Calc. Var. Partial Differential Equations 62(1), Paper No. 9, 34, 2023). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized solutions of coupled Sobolev critical Schrodinger equations with mass subcritical couplings Jianjun, Zhang Xuexiu, Zhong Jinfang, Zhou Analysis of PDEs We are concerned with qualitative properties of positive solutions to the following coupled Sobolev critical Schrödinger equations $$ \begin{cases} -Δu+λ_1 u=μ_1|u|^{2^*-2}u+να|u|^{α-2}|v|^βu ~\hbox{in}~ \R^N,\\ -Δv+λ_2 v=μ_2|v|^{2^*-2}v+νβ|u|^α|v|^{β-2}v ~\hbox{in}~ \R^N \end{cases} $$ subject to the mass constraints $\int_{\mathbb{R}^N}|u|^2 \ud x=a^2$ and $\int_{\mathbb{R}^N}|v|^2 \ud x=b^2$, where, $a>0,\,b>0,\,N=3,4$ and $2^*:=\frac{2N}{N-2}$ is the Sobolev critical exponent. The main purpose of this paper is focused on the mass mixed case, i. e., $ α>1,β>1,α+β<2+\frac{4}{N}$. For some suitable small $ν>0$, we show that the above system admits two positive solutions, one of which is a local minimizer, and another one is a mountain pass solution. Moreover, as $ν\to0^+$, asymptotic behaviors of solutions are also considered. Our result gives an affirmative answer to a Soave's type open problem raised by Bartsch {\it et al.} (Calc. Var. Partial Differential Equations 62(1), Paper No. 9, 34, 2023). |
| title | Normalized solutions of coupled Sobolev critical Schrodinger equations with mass subcritical couplings |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.13163 |