Normalized solutions for a class of gradient-type Schrödinger systems under Neumann boundary conditions in bounded domains

We investigate the existence of normalized solutions to the gradient-type Schr\"odinger system \begin{equation*} \begin{cases} -\Delta u+ V_1(x)u+\lambda u= uv^2 & \text{ in } \Omega,\\ -\Delta v+ V_2(x)v+\lambda v= u^2v & \text{ in } \Omega %\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0 \, & \text{ on } \partial \Omega \end{cases} \end{equation*} subject to the mass constraint $\int_{\Omega}\left(|u|^2+|v|^2 \right)dx=a>0$ and Neumann boundary conditions, where $\Omega\subset \mathbb{R}^3$ is a smooth bounded domain, each $V_i$ is continuous, and $\lambda$ 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 $\mathbb{R}^3_+$.