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 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 ${\int }_{\mathrm{\Omega }}(\mathrm{\mid }u{\mathrm{\mid }}^2+\mathrm{\mid }v{\mathrm{\mid }}^2)dx=a>0$ and Neumann boundary conditions, where $\mathrm{\Omega }\subset {\mathbb{R}}^3$ 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 ${\mathbb{R}}_{+}^3$.