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

Abstract

We are concerned with the existence of normalized solutions to the $L^2$-supercritical nonlinear Schrödinger equation on a noncompact metric graph $G$, \[ \begin{cases} -u''+W(x)u+λu=χ(x)|u|^{p-2}u, & \text{on every edge } e \text{ of } G,\\[2mm] \displaystyle\sum_{e\succ v}u'_e(v)=0, & \text{at every vertex } v\in V, \end{cases} \] under the mass constraint ${\int }_G|u{|}^{2}dx=\mu >0$, 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 ${\int }_1^{\infty }xW(x)dx<\infty$ 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.