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

Abstract

We study the nonlinear Schrödinger 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 ${\phi }_{\omega }(x)$ as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on $V(x)$. Moreover, for $V(x)=-\frac{\beta }{x^{\alpha }}$, in the supercritical case, we prove that the standing waves $e^{i\omega t}{\phi }_{\omega }(x)$ 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.