Solutions with prescribed mass for $L^2$-supercritical NLS equations under Neumann boundary conditions

In this paper, we investigate the following nonlinear Schr\"odinger equation with Neumann boundary conditions: \begin{equation*} \begin{cases} -\Delta u+ \lambda u= f(u) & {\rm in} \,~ \Omega,\\ \displaystyle\frac{\partial u}{\partial \nu}=0 \, &{\rm on}\,~\partial \Omega \end{cases} \end{equation*} coupled with a constraint condition: \begin{equation*} \int_{\Omega}|u|^2 dx=c, \end{equation*} where $\Omega\subset \mathbb{R}^N(N\ge3)$ denotes a smooth bounded domain, $\nu$ represents the unit outer normal vector to $\partial \Omega$, $c$ is a positive constant, and $\lambda$ acts as a Lagrange multiplier. When the nonlinearity $f$ exhibits a general mass supercritical growth at infinity, we establish the existence of normalized solutions, which are not necessarily positive solutions and can be characterized as mountain pass type critical points of the associated constraint functional. Our approach provides a uniform treatment of various nonlinearities, including cases such as $f(u)=|u|^{p-2}u$, $|u|^{q-2}u+ |u|^{p-2}u$, and $-|u|^{q-2}u+|u|^{p-2}u$, where $2<q<2+\frac{4}{N}<p< 2^*$. The result is obtained through a combination of a minimax principle with Morse index information for constrained functionals and a novel blow-up analysis for the NLS equation under Neumann boundary conditions.