AI 中文总结
研究有界区域中一类梯度型薛定谔系统在 Neumann 边界条件和质量约束下归一化解的存在性，运用含 Morse 指标信息的极小极大原理及精细爆破分析等方法，建立了山路型非平凡归一化解的存在性。
AI 中文摘要
我们研究了梯度型薛定谔系统在质量约束和 Neumann 边界条件下归一化解的存在性。通过运用包含 Morse 指标信息的极小极大原理，结合适用于此类梯度型系统的精细爆破分析以及相关极限系统在\(\mathbb{R}^3\)和\(\mathbb{R}^3_+\)中有限 Morse 指标解的新 Liouville 型定理，建立了山路型非平凡归一化解的存在性。
英文摘要
We investigate the existence of normalized solutions to the gradient-type Schrödinger system \begin{equation*} \begin{cases} -Δu+ V_1(x)u+λu= uv^2 & \text{ in } Ω,\\ -Δv+ V_2(x)v+λv= u^2v & \text{ in } Ω%\frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0 \, & \text{ on } \partial Ω\end{cases} \end{equation*} subject to the mass constraint $\int_Ω\left(|u|^2+|v|^2 \right)dx=a>0$ and Neumann boundary conditions, where $Ω\subset \mathbb{R}^3$ is a smooth bounded domain, each $V_i$ is continuous, and $λ$ is a Lagrange multiplier. Applying a minimax principle that incorporates Morse index information, we establish the existence of nontrivial normalized solutions of mountain pass type. The proof is based on a refined blow-up analysis adapted to such gradient-type systems, together with new Liouville-type theorems for finite Morse index solutions of the associated limit systems in $\mathbb{R}^3$ and $\mathbb{R}^3_+$.
