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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.13908 |
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| _version_ | 1866917431447912448 |
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| author | Wu, Chengcheng |
| author_facet | Wu, Chengcheng |
| contents | We investigate the existence of normalized ground states to the system of coupled Schrödinger equations: \begin{equation}\label{eq:0.1}
\begin{cases}
-Δu_1 + λ_1 u_1 = μ_1 |u_1|^{p_1-2}u_1 + βr_1|u_1|^{r_1-2}u_1|u_2|^{r_2} & \text{ in } \mathbb{R}^{3},
-Δu_2 + λ_2 u_2 = μ_2|u_2|^{p_2-2}u_2 + βr_2|u_1|^{r_1}|u_2|^{r_2-2}u_2 & \text{ in } \mathbb{R}^3,
\end{cases}
\end{equation} subject to the constraints $\mathcal{S}_{a_1} \times \mathcal{S}_{a_2} = \{(u_1 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_1^2 dx = a_1^2\} \times \{(u_2 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_2^2 dx = a_2^2\}$, where $μ_1, μ_2 > 0$, $r_1, r_2 > 1$, and $β\geq 0$. Our focus is on the coupled mass super-critical case, specifically, $$\frac{10}{3} < p_1, p_2, r_1 + r_2 < 2^* = 6.$$ We demonstrate that there exists a $\tildeβ \geq 0$ such that equation (\ref{eq:0.1}) admits positive, radially symmetric, normalized ground state solutions when $β> \tildeβ$. Furthermore, this result can be generalized to systems with an arbitrary number of components, and the corresponding standing wave is orbitally unstable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13908 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized grounded states for a coupled nonlinear schrödinger system on $\mathbb{R}^3$ Wu, Chengcheng Analysis of PDEs We investigate the existence of normalized ground states to the system of coupled Schrödinger equations: \begin{equation}\label{eq:0.1} \begin{cases} -Δu_1 + λ_1 u_1 = μ_1 |u_1|^{p_1-2}u_1 + βr_1|u_1|^{r_1-2}u_1|u_2|^{r_2} & \text{ in } \mathbb{R}^{3}, -Δu_2 + λ_2 u_2 = μ_2|u_2|^{p_2-2}u_2 + βr_2|u_1|^{r_1}|u_2|^{r_2-2}u_2 & \text{ in } \mathbb{R}^3, \end{cases} \end{equation} subject to the constraints $\mathcal{S}_{a_1} \times \mathcal{S}_{a_2} = \{(u_1 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_1^2 dx = a_1^2\} \times \{(u_2 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_2^2 dx = a_2^2\}$, where $μ_1, μ_2 > 0$, $r_1, r_2 > 1$, and $β\geq 0$. Our focus is on the coupled mass super-critical case, specifically, $$\frac{10}{3} < p_1, p_2, r_1 + r_2 < 2^* = 6.$$ We demonstrate that there exists a $\tildeβ \geq 0$ such that equation (\ref{eq:0.1}) admits positive, radially symmetric, normalized ground state solutions when $β> \tildeβ$. Furthermore, this result can be generalized to systems with an arbitrary number of components, and the corresponding standing wave is orbitally unstable. |
| title | Normalized grounded states for a coupled nonlinear schrödinger system on $\mathbb{R}^3$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.13908 |