Standing waves of the quintic NLS equation on the tadpole graph

Abstract

AbstractThe 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 $$\omega \in (-\infty ,0)$$ ω ∈ ( - ∞ , 0 ) is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $$L^6$$ L 6 . The set of standing waves includes the set of ground states, which are the global minimizers of the energy at constant mass ($$L^2$$ 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 $$\omega \in (-\infty ,0)$$ ω ∈ ( - ∞ , 0 ) and correspond to a bigger interval of masses. It is proven that there exist critical frequencies $$\omega _1$$ ω 1 and $$\omega _0$$ ω 0 with $$-\infty< \omega _1< \omega _0 < 0$$ - ∞ < ω 1 < ω 0 < 0 such that the standing waves are the ground state for $$\omega \in [\omega _0,0)$$ ω ∈ [ ω 0 , 0 ) , local constrained minima of the energy for $$\omega \in (\omega _1,\omega _0)$$ ω ∈ ( ω 1 , ω 0 ) and saddle points of the energy at constant mass for $$\omega \in (-\infty ,\omega _1)$$ ω ∈ ( - ∞ , ω 1 ) . Proofs make use of the variational methods and the analytical theory for differential equations.