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}- Δ_α u + ω u=w u+β u |u|p - 2&on ~ R^2;\\- Δ w=2 π |u|^2&on ~ R^2;\\\|u\|L^2^2 = c,{cases}{equation*} where p > 2 ; α, β ∈ R ; c > 0 and - Δ_α denotes the Laplacian of point interaction with s-wave scattering length (- 2 π α)- 1 , the unknowns being u R² → C , w R² → 0, ∞ and the Lagrange multiplier ω ∈ 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 ψ' (t)=- Δ_α ψ (t)-(log |·| |ψ (t)|^2)ψ (t)-βψ (t)|ψ (t)|p - 2.{equation*}