Constrained energy minimization and orbital stability for the NLS equation on a star graph

On a star graph GG, we consider a nonlinear Schrödinger equation with focusing nonlinearity of power type and an attractive Dirac's delta potential located at the vertex. The equation can be formally written as i∂tΨ(t)=−ΔΨ(t)−|Ψ(t)|2μΨ(t)+αδ0Ψ(t)i∂tΨ(t)=−ΔΨ(t)−|Ψ(t)|2μΨ(t)+αδ0Ψ(t), where the strength α of the vertex interaction is negative and the wave function Ψ   is supposed to be continuous at the vertex. The values of the mass and energy functionals are conserved by the flow. We show that for 0<μ⩽20<μ⩽2 the energy at fixed mass is bounded from below and that for every mass m   below a critical mass m⁎m⁎ it attains its minimum value at a certain Ψˆm∈H1(G).Moreover, the set of minimizers has the structure M={eiθΨˆm,θ∈R}. Correspondingly, for every m