Standing waves of the quintic NLS equation on the tadpole graph

The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schr"{o}dinger equation with quintic power nonlinearity equipped with the Neumann-Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency $\omega \in (-\mathrm{\infty },0)$ is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $L^6$. The set of minimizers includes the set of ground states of the system, which are the global minimizers of the energy at constant mass ($L^2$-norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every $\omega \in (-\mathrm{\infty },0)$ and correspond to a bigger interval of masses. It is shown that there exist critical frequencies ${\omega }_0$ and ${\omega }_1$ such that the standing waves are the ground states for $\omega \in [{\omega }_0,0)$, local minimizers of the energy at constant mass for $\omega \in ({\omega }_1,{\omega }_0)$, and saddle points of the energy at constant mass for $\omega \in (-\mathrm{\infty },{\omega }_1)$. Proofs make use of both the variational methods and the analytical theory for differential equations.