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 $$\varphi _{\omega }(x)$$ 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.