Abstract
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 $$\omega \in (-\infty ,0)$$
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.
Orbital instability of standing waves for NLS equation on star graphs