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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 Schrödinger equation with quintic power nonlinearity equipped with the Neumann-Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency $ω\in (-\infty,0)$ 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 $ω\in (-\infty,0)$ and correspond to a bigger interval of masses. It is shown that there exist critical frequencies $ω_0$ and $ω_1$ such that the standing waves are the ground states for $ω\in [ω_0,0)$, local minimizers of the energy at constant mass for $ω\in (ω_1,ω_0)$, and saddle points of the energy at constant mass for $ω\in (-\infty,ω_1)$. Proofs make use of both the variational methods and the analytical theory for differential equations.