Normalized ground states for a doubly nonlinear Schrödinger equation on periodic metric graphs

Xiaoguang Li; Xiaoguang Li

doi:10.3934/era.2024189

Electronic Research Archive

2024, Volume 32, Issue 7: 4199-4217. doi: 10.3934/era.2024189

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Research article

Normalized ground states for a doubly nonlinear Schrödinger equation on periodic metric graphs

Xiaoguang Li ^,

College of Sciences, University of Shanghai for Science and Technology, Shanghai 200093, China

Received: 06 May 2024 Revised: 16 June 2024 Accepted: 25 June 2024 Published: 01 July 2024

We investigate the existence of ground states for a class of Schrödinger equations with both a standard power nonlinearity and delta nonlinearity concentrated at finite vertices of the periodic metric graphs $ G $. Using variational methods, if $ \alpha > 0 $ and the standard nonlinearity power is $ L^{2}- $subcritical, we establish the existence of ground states for every mass and every periodic graph. If $ \alpha < 0 $ and the standard nonlinearity power is $ L^{2}- $critical, we show that two types of topological structures on $ G $ will prevent the existence of ground states. Furthermore, for graphs that do not satisfy these two types of topological structures, ground states exist when the given mass belongs to an appropriate range and the parameter $ \left | \alpha \right| $ is small enough.
- ground states,
- point-wise nonlinearity,
- periodic metric graph
Citation: Xiaoguang Li. Normalized ground states for a doubly nonlinear Schrödinger equation on periodic metric graphs[J]. Electronic Research Archive, 2024, 32(7): 4199-4217. doi: 10.3934/era.2024189

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Abstract

We investigate the existence of ground states for a class of Schrödinger equations with both a standard power nonlinearity and delta nonlinearity concentrated at finite vertices of the periodic metric graphs $ G $. Using variational methods, if $ \alpha > 0 $ and the standard nonlinearity power is $ L^{2}- $subcritical, we establish the existence of ground states for every mass and every periodic graph. If $ \alpha < 0 $ and the standard nonlinearity power is $ L^{2}- $critical, we show that two types of topological structures on $ G $ will prevent the existence of ground states. Furthermore, for graphs that do not satisfy these two types of topological structures, ground states exist when the given mass belongs to an appropriate range and the parameter $ \left | \alpha \right| $ is small enough.

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