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∈(-∞,0)$$\omega \in (-\infty ,0)$$ is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in L6$$L^6$$. The set of standing waves includes the set of ground states, which are the global minimizers of the energy at constant mass (L2$$L^2$$-norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every ω∈(-∞,0)$$\omega \in (-\infty ,0)$$ and correspond to a bigger interval of masses. It is proven that there exist critical frequencies ω1$$\omega _1$$ and ω0$$\omega _0$$ with -∞<ω1<ω0<0$$-\infty< \omega _1< \omega _0 < 0$$ such that the standing waves are the ground state for ω∈[ω0,0)$$\omega \in [\omega _0,0)$$, local constrained minima of the energy for ω∈(ω1,ω0)$$\omega \in (\omega _1,\omega _0)$$ and saddle points of the energy at constant mass for ω∈(-∞,ω1)$$\omega \in (-\infty ,\omega _1)$$. Proofs make use of the variational methods and the analytical theory for differential equations.