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Constant sign and
sign changing NLS ground states on noncompact metric
graphs
Colette De Coster, Simone Dovetta, Damien Galant, Enrico Serra and Christophe Troestler |
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Vol. 19 (2026), No. 2, 203–240
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Abstract |
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We investigate existence and nonexistence of action ground states and nodal action ground states for the nonlinear Schrödinger equation on noncompact metric graphs with mixed homogeneous Kirchhoff and Dirichlet boundary conditions. We first obtain abstract sufficient conditions for existence, typical of problems with lack of compactness, in terms of “levels at infinity” for the action functional associated with the problems. Then we analyze in detail two relevant classes of graphs. For noncompact graphs with at least one half-line, we detect purely topological sharp conditions preventing the existence of ground states or of nodal ground states. We also investigate analogous conditions of metrical nature. The negative results are complemented by several sufficient conditions to ensure existence, either of topological or metrical nature, or a combination of the two. For graphs with infinitely many edges, all bounded, we focus on periodic graphs and infinite trees. In these cases, our results completely describe the phenomenology. Furthermore, we study nodal domains and nodal sets of nodal ground states and we show that the situation on graphs can be totally different from that on domains of . |
Keywords
nonlinear Schrödinger, ground states, nodal solutions,
least action, constrained minimization
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Mathematical Subject Classification
Primary: 35Q55, 35R02, 49J40, 58E30
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Milestones
Received: 25 September 2023
Revised: 25 September 2024
Accepted: 1 February 2025
Published: 22 January 2026
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Authors |
| Colette De Coster | ||
| Université Polytechnique
Hauts-de-France INSA Hauts-de-France CERAMATHS - Laboratoire de Matériaux Céramiques et de Mathématiques F-59313 Valenciennes France |
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| Simone Dovetta | ||
| Dipartimento di Scienze Matematiche
“G. L. Lagrange” Politecnico di Torino Torino Italy |
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| Damien Galant | ||
| Département de Mathématiques Université de Mons Mons Belgium |
Université Polytechnique
Hauts-de-France INSA Hauts-de-France CERAMATHS - Laboratoire de Matériaux Céramiques et de Mathématiques F-59313 Valenciennes France |
Department of Mathematics Brown University Providence, RI United States |
| Enrico Serra | ||
| Dipartimento di Scienze Matematiche
“G. L. Lagrange” Politecnico di Torino Torino Italy |
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| Christophe Troestler | ||
| Département de Mathématiques Université de Mons Mons Belgium |
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| © 2026 MSP (Mathematical Sciences Publishers). |
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