Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

Failure of scattering for the NLSE with a point interaction in dimension two and three

“…We consider the nonlinear Schrödinger equation i ∂ t ψ = H α ψ + F(ψ), (1.1) where H α is the family of self-adjoint operators known as point interactions (see later) and F(ψ) = ±|ψ| p−1 ψ . This equation has been recently considered, in dimension two and three, by several authors as regards its well posedness in different functional frameworks ( [8,14]), existence and stability of standing waves ( [1,2]) and blow-up ( [13]). When α = +∞ the operator H α coincides with the Laplacian and (1.1) is the standard nonlinear Schrödinger (NLS) equation.…”

“…The asymptotics(3.14) follows from lemma 3.1, since the constraint2 2−p < 3, needed for n = 3, is satisfied for p ∈ (1, 4/3). Next we prove(3.13).…”

Well–posedness of the three–dimensional NLS equation with sphere–concentrated nonlinearity

“…[8-10, 13, 51]). In 2D and 3D, on the contrary, first well-posedness results have been obtained in [22], whereas standing waves have been discussed in [1,2,38].…”

A general review on the NLS equation with point-concentrated nonlinearity

“…Results on (3), which are not presented in this review, are discussed by [17][18][19][20][21][22][23][24][25][26] in dimension one and by [27][28][29][30][31][32] in dimensions two and three (while [33] addresses the circle, [34] addresses the half-line and [35][36][37] concern some first studies on a mixed model between ( 2) and ( 3)).…”