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) is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in L6. The set of standing waves includes the set of ground states, which are the global minimizers of the energy at constant mass (L2-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) and correspond to a bigger interval of masses. It is proven that there exist critical frequencies ω1 and ω with - ∞< ω1< ω< 0 such that the standing waves are the ground state for ω∈ [ω, 0) , local constrained minima of the energy for ω∈ (ω1, ω) and saddle points of the energy at constant mass for ω∈ (- ∞, ω1). Proofs make use of the variational methods and the analytical theory for differential equations.