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

Abstract

On a star graph $\mathcal{G}$, 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 $i{\partial }_t\Psi (t)=-\mathrm{\Delta }\Psi (t)-|\Psi (t){|}^{2\mu }\Psi (t)+\alpha {\delta }_0\Psi (t)$, where the strength $\alpha$ of the vertex interaction is negative and the wave function $\Psi$ 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 $0<\mu ⩽2$ the energy at fixed mass is bounded from below and that for every mass $m$ below a critical mass $m^{⁎}$ it attains its minimum value at a certain ${\hat{\Psi }}_m\in H^1(\mathcal{G})$.

Moreover, the set of minimizers has the structure $\mathcal{M}=\{e^{i\theta }{\hat{\Psi }}_m,\theta \in \mathbb{R}\}$. Correspondingly, for every $m<m^{⁎}$ there exists a unique $\omega =\omega (m)$ such that the standing wave ${\hat{\Psi }}_{\omega }{e}^{i\omega t}$ is orbitally stable. To prove the above results we adapt the concentration-compactness method to the case of a star graph. This is nontrivial due to the lack of translational symmetry of the set supporting the dynamics, i.e. the graph. This affects in an essential way the proof and the statement of concentration-compactness lemma and its application to minimization of constrained energy. The existence of a mass threshold comes from the instability of the system in the free (or Kirchhoff's) case, that in our setting corresponds to $\alpha =0$.