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

1. Problem Statement

The paper investigates the existence and multiplicity of prescribed mass solutions (also known as bound states) for the stationary Nonlinear Schrödinger (NLS) equation set on a compact metric graph $G$. The equation is given by:
$$\begin{cases} -u'' + \lambda u = |u|^{p-2}u & \text{on every edge } e \in E, \\ \sum_{e \succ v} u'_e(v) = 0 & \text{at every vertex } v \in V \quad (\text{Kirchhoff condition}), \end{cases}$$
subject to the mass constraint:
$$\int_G |u|^2 dx = \mu > 0.$$
The study focuses on the $L^2$-supercritical regime, where the nonlinearity exponent satisfies $p > 6$.

Key Challenges:

• Supercriticality: Unlike the subcritical case ($p < 6$), the energy functional is unbounded from below on the mass constraint, making standard minimization techniques inapplicable.
• Compactness: On compact graphs, the energy functional always admits a trivial constant solution. The goal is to find non-constant solutions.
• Sign-Changing Solutions: While positive solutions were previously established (Jeanjean et al., [14]), the existence of sign-changing (nodal) solutions with prescribed mass in the supercritical regime on compact graphs was an open problem.
• Constraint Handling: Standard techniques for finding sign-changing solutions (e.g., Pohozaev manifolds or symmetry arguments in $\mathbb{R}^N$) fail on bounded domains and compact graphs due to the lack of scaling invariance and the specific nature of the mass constraint.

2. Methodology

The authors employ a combination of variational methods, min-max theory, and gradient flow techniques adapted to the mass constraint.

A. Variational Framework

The problem is formulated as finding critical points of the energy functional:
$$E(u, G) = \frac{1}{2} \int_G |u'|^2 dx - \frac{1}{p} \int_G |u|^p dx$$
constrained on the $L^2$-sphere $H^1_\mu(G) = \{ u \in H^1(G) : \|u\|_{L^2}^2 = \mu \}$.

B. The Working Set and Boundedness

To handle the lack of coercivity, the authors restrict their analysis to a bounded subset of the constraint:
$$B^\mu_\rho = \left\{ u \in H^1_\mu(G) : \int_G |u'|^2 dx < \rho \right\}.$$
They prove that for sufficiently small mass $\mu$, the Palais-Smale (PS) condition holds on the closure of this set, and critical points found within this set are valid solutions without needing additional constraints.

C. Min-Max Structure and Linking

The authors construct a min-max structure using a "linking" argument between subspaces defined by the eigenfunctions of the linear operator $-\frac{d^2}{dx^2}$ with Kirchhoff conditions.

• Let $0 = \lambda_1 < \lambda_2 \le \lambda_3 \dots$ be the eigenvalues.
• They define subspaces $S_k = \text{span}\{\phi_1, \dots, \phi_k\} \cap H^1_\mu(G)$ and their orthogonal complements $S_k^\perp$.
• A specific compact set $Q_k$ is constructed to link $S_{k-1}$ with $S_{k-1}^\perp$.
• The min-max level $c_k$ is defined as the infimum over all such linking sets of the supremum of the energy.

D. Invariant Sets and Gradient Flow

A major technical novelty is the development of a framework to find sign-changing solutions under a mass constraint.

• Challenge: The classical Brezis-Martin result (used to prove invariance of sets like $\{u \ge 0\}$ under gradient flow) cannot be directly applied because the Lagrange multiplier $\lambda_u$ depends on $u$ in the constrained case.
• Solution: The authors extend the Brezis-Martin result to an abstract setting involving constrained manifolds (Proposition 4.5). They prove that the sets of functions close to non-negative or non-positive functions, denoted $(\pm P)_\nu$, are invariant under a specific negative gradient flow $\eta(t, u)$ on the constraint $H^1_\mu(G)$.
• This invariance ensures that if a flow starts in the "sign-changing" region (outside the invariant sets), it cannot cross into the region of constant sign, guaranteeing that the critical point found is indeed sign-changing.

3. Key Contributions

1. First Multiplicity Result for Sign-Changing Solutions: This is the first paper to prove the existence of multiple sign-changing solutions with prescribed mass for the $L^2$-supercritical NLS on compact metric graphs. Previous results were limited to non-compact graphs or subcritical regimes.
2. General Framework for Constrained Flows: The paper provides a general abstract framework (Propositions 4.5, 4.6, and Lemma 4.7) to establish the invariance of subsets under gradient flows on manifolds defined by mass constraints. This overcomes the difficulty of the variable Lagrange multiplier and is applicable to other constrained problems on bounded domains.
3. Bifurcation Analysis: The authors characterize the bifurcation points of the problem as the mass $\mu \to 0$.

4. Main Results

Theorem 1.1 (Existence of Sign-Changing Solutions)

For any compact metric graph $G$ and $p > 6$, there exists a threshold $\mu_2 > 0$ such that for any mass $0 < \mu < \mu_2$:

• The energy functional $E(\cdot, G)$ possesses at least two pairs of non-constant critical points $\{\pm u_1, \pm u_2\}$.
• $u_1$ is a positive solution (mountain pass level).
• $u_2$ is a sign-changing solution.

Theorem 1.3 (Arbitrary Multiplicity)

For any integer $j \in \mathbb{N}$, there exists a threshold $\mu_j > 0$ such that for $0 < \mu < \mu_j$, the functional has at least $j$ pairs of non-constant critical points.

• One pair is positive/negative.
• The remaining $j-1$ pairs are sign-changing.
• The distinctness of these solutions is guaranteed by estimating their energy levels, which correspond to different eigenvalues of the linearized operator.

Theorem 1.4 (Bifurcation Phenomena)

As the mass $\mu \to 0$, the solutions obtained bifurcate from the eigenvalues of the linear operator. Specifically, the set of bifurcation points for the system is exactly the set of all eigenvalues $\{\lambda_j(G) : j \in \mathbb{N}\}$.

• This result is significant because it shows that every eigenvalue (even those with multiplicity $>1$) acts as a bifurcation point for sign-changing solutions in the supercritical regime, extending classical bifurcation theory which typically only guarantees bifurcation from simple eigenvalues.

5. Significance

• Mathematical Advancement: The paper resolves a long-standing open problem regarding the multiplicity of solutions in the supercritical regime on compact structures. It bridges the gap between unconstrained sign-changing solution theory and constrained mass-preserving problems.
• Physical Relevance: The results are relevant for modeling optical pulse propagation in nonlinear optical networks (e.g., T-junctions, X-junctions) and matter waves in Bose-Einstein condensates on branched structures, where the mass (particle number) is conserved and the nonlinearity is strong.
• Methodological Impact: The developed technique for proving the invariance of sets under constrained gradient flows is a powerful tool that can be transposed to other variational problems on bounded domains where the mass constraint prevents the use of standard scaling arguments.

In summary, this work establishes a robust theoretical foundation for the existence of complex, sign-changing wave patterns in supercritical nonlinear systems on compact networks, utilizing a novel blend of topological linking and constrained dynamical systems.