Sign-changing prescribed mass solutions for $L^2$-supercritical NLS on compact metric graphs

TL;DR

This work proves the existence of sign-changing, prescribed-mass bound states for the $L^2$-supercritical nonlinear Schrödinger equation on a compact metric graph when $p>6$. The authors develop a novel linking structure within the mass-constrained space $H^1_\mu(\mathcal{G})$ and employ a gradient-flow on the constraint, together with a constrained Brézis–Martin framework, to produce multiple sign-changing critical points of the energy $E(u,\mathcal{G})$. They show, for small mass $\mu$, at least two non-constant pairs $\pm u_1$ and $\pm u_2$ (with $u_1$ positive and $u_2$ sign-changing), and, more generally, that one can obtain arbitrarily many sign-changing states as $\mu$ is taken smaller ($\mu_j$-thresholds). A bifurcation analysis reveals that every eigenvalue $\lambda_j(\mathcal{G})$ of the linear Kirchhoff operator is a bifurcation point for the constrained problem, with branches of solutions emanating from each eigenmode in the $\mu\to0$ limit. The methods are robust and adaptable to bounded domains, offering a general strategy for constrained sign-changing bound states in $L^2$-supercritical settings.

Abstract

This paper is devoted to the existence of multiple sign-changing solutions of prescribed mass for a mass-supercritical nonlinear Schrödinger equation set on a compact metric graph. In particular, we obtain, in the supercritical mass regime, the first multiplicity result for prescribed mass solutions on compact metric graphs. As a byproduct, we prove that any eigenvalue of the associated linear operator is a bifurcation point. Our approach relies on the introduction a new kind of link and on the use of gradient flow techniques on a constraint. It can be transposed to other problems posed on a bounded domain.

Theorems & Definitions (43)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Corollary 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof