Classification and stability of positive solutions to the NLS equation on the T -metric graph

Abstract

Abstract Given λ > 0 and p > 2, we present a complete classification of the positive H 1-solutions of the equation − u ′ ′ + λ u = | u | p − 2 u on the T -metric graph (consisting of two unbounded edges and a terminal edge of length ℓ > 0 , all joined together at a single vertex). This study implies, in particular, the uniqueness of action ground states. Moreover, for p ∼ 6 − , the notions of action and energy ground states do not coincide and energy ground states are not unique. In the L 2-supercritical case p > 6, we prove that, for λ ∼ 0 + and λ ∼ + ∞ , action ground states are orbitally unstable for the flow generated by the associated time-dependent NLS equation i ∂ t u + ∂ x x 2 u + | u | p − 2 u = 0 . Finally, we provide numerical evidence of the uniqueness of energy ground states for p ∼ 2 + and of the existence of both stable and unstable action ground states for p ∼ 6 .