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

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