Multiple positive solutions with prescribed masses for a coupled Schrödinger system: mass mixed and Sobolev critical coupled case
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2026
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| _version_ | 1866910169679527936 |
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| author | Guo, Qing He, Qihan Shuai, Wei Zhong, Xuexiu |
| author_facet | Guo, Qing He, Qihan Shuai, Wei Zhong, Xuexiu |
| contents | The aim of this paper is to establish multiple positive normalized solutions $(u,v,λ_1,λ_2)\in H^1(\mathbb{R}^N,\mathbb{R}^2)\times \mathbb{R}^2$ to the following coupled Schrödinger system involving Sobolev critical exponent: $$ \begin{cases} -Δu+λ_1 u=μ_1|u|^{p-2}u+να|u|^{α-2}u|v|^β, x\in \mathbb{R}^N,\\ -Δv+λ_2 v=μ_2|v|^{q-2}v+νβ|v|^{β-2}v|u|^α, x\in \mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2\mathrm{d}x=a, \int_{\mathbb{R}^N}|v|^2\mathrm{d}x=b, \end{cases} N\geq 3, $$ where $μ_1,μ_2, ν, a, b>0$. We are particularly interested in the mass mixed case that $2<p, q<2+\frac{4}{N}, α>1, β>1$, and $α+β=2^*:=\frac{2N}{N-2}$. For sufficiently small $ν>0$, we demonstrate that the above system admits two positive solutions, one of which serves as a local minimizer, and the other as a mountain pass solution. By developing some new technical lemmas on the interaction estimates, we are managed to resolves Soave's open problem [{\it J. Funct. Anal.}, 2020, Remark 1.1] within the context of the system case. Notably, our existence result holds true for all dimensions $N\geq 3$. Our results also significantly extend the result of Gou and Jeanjean [{\it Nonlinearity}, 2018, Theorem 1.1] to the Sobolev critical coupled case and removing the hypothesis ``either $p,q\leq α+β-\frac{2}{N}$ or $|p-q|\leq \frac{2}{N}$" for $N\geq 5$. Additionally, we also establish a sequence of properties for the local minimizer, including local uniqueness, continuity with respect to the small parameter $ν$, and the limiting profiles for $ν\rightarrow 0^+$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_24438 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiple positive solutions with prescribed masses for a coupled Schrödinger system: mass mixed and Sobolev critical coupled case Guo, Qing He, Qihan Shuai, Wei Zhong, Xuexiu Analysis of PDEs 35Q55, 35J50, 35B33 The aim of this paper is to establish multiple positive normalized solutions $(u,v,λ_1,λ_2)\in H^1(\mathbb{R}^N,\mathbb{R}^2)\times \mathbb{R}^2$ to the following coupled Schrödinger system involving Sobolev critical exponent: $$ \begin{cases} -Δu+λ_1 u=μ_1|u|^{p-2}u+να|u|^{α-2}u|v|^β, x\in \mathbb{R}^N,\\ -Δv+λ_2 v=μ_2|v|^{q-2}v+νβ|v|^{β-2}v|u|^α, x\in \mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2\mathrm{d}x=a, \int_{\mathbb{R}^N}|v|^2\mathrm{d}x=b, \end{cases} N\geq 3, $$ where $μ_1,μ_2, ν, a, b>0$. We are particularly interested in the mass mixed case that $2<p, q<2+\frac{4}{N}, α>1, β>1$, and $α+β=2^*:=\frac{2N}{N-2}$. For sufficiently small $ν>0$, we demonstrate that the above system admits two positive solutions, one of which serves as a local minimizer, and the other as a mountain pass solution. By developing some new technical lemmas on the interaction estimates, we are managed to resolves Soave's open problem [{\it J. Funct. Anal.}, 2020, Remark 1.1] within the context of the system case. Notably, our existence result holds true for all dimensions $N\geq 3$. Our results also significantly extend the result of Gou and Jeanjean [{\it Nonlinearity}, 2018, Theorem 1.1] to the Sobolev critical coupled case and removing the hypothesis ``either $p,q\leq α+β-\frac{2}{N}$ or $|p-q|\leq \frac{2}{N}$" for $N\geq 5$. Additionally, we also establish a sequence of properties for the local minimizer, including local uniqueness, continuity with respect to the small parameter $ν$, and the limiting profiles for $ν\rightarrow 0^+$. |
| title | Multiple positive solutions with prescribed masses for a coupled Schrödinger system: mass mixed and Sobolev critical coupled case |
| topic | Analysis of PDEs 35Q55, 35J50, 35B33 |
| url | https://arxiv.org/abs/2604.24438 |