Trapped focusing cubic ground-state equation ($\Delta P_{\eta }-P_{\eta }-\frac{\eta }{4}|x{|}^2{P}_{\eta }+P_{\eta }^3=0$)

The trapped focusing cubic ground-state equation in two dimensions isA nonzero positive solution follows from minimizing the harmonic-oscillator quadratic form subject to a fixed $L^4$ norm and then rescaling by the positive Lagrange multiplier.

If $P_{\eta }$ solves the trapped focusing cubic ground-state equation, the ansatzsolves the two-dimensional mass-critical focusing nonlinear Schrödinger equation when

The initial conditions $\lambda (-1)=\sqrt{1+\eta }$ and $b(-1)=1$ selectThe scale grows linearly at infinity, yielding a finite critical spacetime norm for the corresponding lens solution.