Ground states of the planar nonlinear Schrödinger–Newton system with a point interaction

We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schrödinger–Newton system with a point interaction: equation*cases- _ u + u=w u+ u |u|^p - 2\\- w=2 |u|²\\\|u\|₋ℂ² = c, casesequation* where p 2 ;, R ; c 0 and - _ denotes the Laplacian of point interaction with s-wave scattering length (- 2) ^- 1, the unknowns being u R² C, w R² 0, and the Lagrange multiplier R. Additionally, we show that critical points of the corresponding constrained energy functional are naturally associated with standing waves of the evolution problem equation*i ' (t) =- _ (t) - (|| | (t) |²) (t) - (t) | (t) |^p - 2. equation*