Instability of ground states for the NLS equation with potential on the star graph

TL;DR

This paper investigates the existence and instability of ground states for the nonlinear Schrödinger equation with potential on a star graph, revealing conditions under which standing waves become orbitally unstable.

Contribution

It establishes the existence of ground states under certain conditions and proves orbital instability of standing waves in the supercritical case with specific potentials.

Findings

Existence of ground states as minimizers under negativity and decay conditions.

Orbital instability of standing waves for supercritical potentials.

Instability results hold for large frequencies and symmetric profiles.

Abstract

We study the nonlinear Schr\"odinger equation with an arbitrary real potential $V(x)\in (L^1+L^\infty)(\Gamma)$ on a star graph $\Gamma$. At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength $-\gamma<0$. We show the existence of ground states $\varphi_{\omega}(x)$ as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on $V(x)$. Moreover, for $V(x)=-\dfrac{\beta}{x^\alpha}$, in the supercritical case, we prove that the standing waves $e^{i\omega t}\varphi_{\omega}(x)$ are orbitally unstable in $H^{1}(\Gamma)$ when $\omega$ is large enough. Analogous result holds for an arbitrary $\gamma\in\mathbb{R}$ when the standing waves have symmetric profile.