Normalized Solutions of the $L^2$-Supercritical NLS Equation on Noncompact Metric Graphs with a Short-Range Potential

We are concerned with the existence of normalized solutions to the $L^2$-supercritical nonlinear Schr"odinger equation on a noncompact metric graph $G$, $$\{\begin{array}{ll}-u^{\mathrm{\prime }\mathrm{\prime }}+W(x)u+\lambda u=\chi (x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,& \text{on every edge }e\text{ of }G,\\ {\displaystyle \sum _{e\succ v}{u}_e^{\mathrm{\prime }}(v)=0,}& \text{at every vertex }v\in V,\end{array}$$ under the mass constraint ${\int }_G\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=\mu >0$, where $\lambda$ arises as a Lagrange multiplier. Here $p>6$, $\chi$ is the characteristic function of the compact core $\mathcal{K}$, so that the nonlinearity is localized, and the potential $W$ is bounded, nonnegative and vanishing at infinity along every unbounded edge, with ${\int }_1^{\mathrm{\infty }}xW(x)dx<\mathrm{\infty }$ on at least one of them. For every $\mu >0$ we obtain a positive solution with $\lambda >0$, arising as a constrained critical point at a strictly positive energy level. Our approach combines a uniform mountain-pass geometry for a family of approximating functionals, the monotonicity trick with Morse index type information, and a blow-up analysis ruling out the divergence of the Lagrange multipliers. A key point is the strict positivity of the multiplier, for which we exhibit a potential of inverse-square decay admitting a zero-energy $L^2$ solution on a half-line. To the best of our knowledge, this is the first existence result for normalized solutions of the $L^2$-supercritical NLS equation on a noncompact metric graph in the presence of an external potential.