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Normalized solutions for the Kirchhoff equation on noncompact metric graphs. (English) Zbl 07396377

This work studies the nonlocal equation \[-\left(a+b\int_{\mathcal{G}}|u^\prime|^2dx \right)u^{\prime\prime}+\lambda u=|u|^{p-2}u,\;\;x\in\mathcal{G}\] with the constraint \[\int_{\mathcal{G}}|u|^2dx=\mu,\] where \(\mathcal{G}\) is a compact metric graph, arising in the study of stationary solution of the model Kirchoff and Carrier for vibrations of an elastic string. The problem has a natural variational formulation, which is used in its study. Detailed background on the problem and on previous results is given in the Introduction. The paper demonstrates the rich structure and complexity of the problem. It first studies the cases in which the graph \(\mathcal{G}\) is either a whole line or a half-line, showing that, depending on the value of \(p\) and of \(\mu\), the equation can have no solution, a unique solution, or two solutions. Further, existence, non-existence and uniqueness results are given for non-compact metric graphs \(\mathcal{G}\) under a general topological assumption, as well as for specific graphs composed of half lines and a terminal edge.

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
47J30 Variational methods involving nonlinear operators
34A09 Implicit ordinary differential equations, differential-algebraic equations
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
35L72 Second-order quasilinear hyperbolic equations
49J40 Variational inequalities
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
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