Standing waves for NLS with delta-interaction on a star graph are strongly unstable for p at least 5 when alpha is positive, and for p greater than 5 above an explicit frequency threshold when alpha is negative.
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Blow-up and strong instability of standing waves for the NLS-$\delta$ equation on a star graph
Standing waves for NLS with delta-interaction on a star graph are strongly unstable for p at least 5 when alpha is positive, and for p greater than 5 above an explicit frequency threshold when alpha is negative.