Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities

Normalized solution to the Schödinger equation with potential and general nonlinear term: Mass super-critical case

“…And the key step is to establish the strict so-called subadditive inequality. In [8], Norihisa Ikoma and Yasuhito Miyamoto applied the standard concentration compactness arguments, which is due to Lions [14,15], to make the first progress on this direction. Recently, under different assumptions, Zhong and Zou [28] present a new approach based on iteration, to establish the strict sub-additive inequality, which can greatly simplify the discussion process in the traditional sense.…”

Normalized solutions for nonlinear Schrödinger systems with linear couples

“…is also studied under some suitable assumptions on V (x) by some researchers. The mass-subcritical case, we refer to Norihisa Ikoma and Yasuhito Miyamoto [19], see also Zhong and Zou [31]. While the mass super-critical case we refer to Bartsch et al [7].…”

Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction

“…Although the condition (V3) is not necessary for the scalar case or some systems (see e.g. [4], [16]), we need the condition (V3) to overcome the difficulty to establish the strict subadditivity (see Proposition 3) for our system with three wave interaction by using the coupled rearrangement technique. We note that Bhattarai [4] and studied L 2 -constrained variational problems with local and nonlocal nonlinearities based on new arguments to establish the strict subadditivity.…”