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) ∈ (L 1 +L ∞ )(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) = − β 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 ∈ R when the standing waves have symmetric profile.