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$$L^6$$. …