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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2410.22614 |
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| _version_ | 1866913567882608640 |
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| author | Chen, Zhijie Zhao, Hanqing |
| author_facet | Chen, Zhijie Zhao, Hanqing |
| contents | We study the weakly coupled nonlinear Schrödinger system \begin{equation*} \begin{cases} -Δu_1 = μ_1 u_1^{p} +βu_1^{\frac{p-1}{2}} u_2^{\frac{p+1}{2}}\text{ in } Ω,\\ -Δu_2 = μ_2 u_2^{p} +βu_2^{\frac{p-1}{2}}u_1^{\frac{p+1}{2}} \text{ in } Ω,\\ u_1,u_2>0\quad\text{in }\;Ω;\quad u_1=u_2=0 \quad\text { on } \;\partialΩ, \end{cases} \end{equation*} where $p>1, μ_1, μ_2, β>0$ and $Ω$ is a smooth bounded domain in $\mathbb{R}^2$. Under the natural condition that holds automatically for all positive solutions in star-shaped domains \begin{align*}
p\int_Ω|\nabla u_{1,p}|^2+|\nabla u_{2,p}|^2 dx \leq C, \end{align*} we give a complete description of the concentration phenomena of positive solutions $(u_{1,p},u_{2,p})$ as $p\rightarrow+\infty$, including the $L^{\infty}$-norm quantization $\|u_{k,p}\|_{L^\infty(Ω)}\to \sqrt{e}$ for $k=1,2$, the energy quantization $p\int_Ω|\nabla u_{1,p}|^2+|\nabla u_{2,p}|^2dx\to 8nπe $ with $n\in\mathbb{N}_{\geq 2}$, and so on. In particular, we show that the ``local mass'' contributed by each concentration point must be one of $\{(8π,8π), (8π,0),(0,8π)\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22614 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concentration phenomena of positive solutions to weakly coupled Schrödinger systems with large exponents in dimension two Chen, Zhijie Zhao, Hanqing Analysis of PDEs We study the weakly coupled nonlinear Schrödinger system \begin{equation*} \begin{cases} -Δu_1 = μ_1 u_1^{p} +βu_1^{\frac{p-1}{2}} u_2^{\frac{p+1}{2}}\text{ in } Ω,\\ -Δu_2 = μ_2 u_2^{p} +βu_2^{\frac{p-1}{2}}u_1^{\frac{p+1}{2}} \text{ in } Ω,\\ u_1,u_2>0\quad\text{in }\;Ω;\quad u_1=u_2=0 \quad\text { on } \;\partialΩ, \end{cases} \end{equation*} where $p>1, μ_1, μ_2, β>0$ and $Ω$ is a smooth bounded domain in $\mathbb{R}^2$. Under the natural condition that holds automatically for all positive solutions in star-shaped domains \begin{align*} p\int_Ω|\nabla u_{1,p}|^2+|\nabla u_{2,p}|^2 dx \leq C, \end{align*} we give a complete description of the concentration phenomena of positive solutions $(u_{1,p},u_{2,p})$ as $p\rightarrow+\infty$, including the $L^{\infty}$-norm quantization $\|u_{k,p}\|_{L^\infty(Ω)}\to \sqrt{e}$ for $k=1,2$, the energy quantization $p\int_Ω|\nabla u_{1,p}|^2+|\nabla u_{2,p}|^2dx\to 8nπe $ with $n\in\mathbb{N}_{\geq 2}$, and so on. In particular, we show that the ``local mass'' contributed by each concentration point must be one of $\{(8π,8π), (8π,0),(0,8π)\}$. |
| title | Concentration phenomena of positive solutions to weakly coupled Schrödinger systems with large exponents in dimension two |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.22614 |