Doubly nonlinear Schr\"odinger normalized ground states on 2D grids: existence results and singular limits

TL;DR

This paper studies the existence and limits of ground states for doubly nonlinear Schr"odinger equations on 2D grids, revealing conditions for existence and showing convergence to continuum models as grid edges shrink.

Contribution

It provides new existence criteria and demonstrates strong convergence of piecewise-affine states to continuum ground states in doubly nonlinear Schr"odinger models.

Findings

Existence and non-existence conditions depend on nonlinearity powers and vertex structure.

Ground states on grids converge strongly to continuum models as grid edges tend to zero.

Convergence holds for models with standard and singular nonlinearities.

Abstract

We investigate the existence and the singular limit of normalized ground states for focusing doubly nonlinear Schr\"odinger equations with both standard and concentrated nonlinearities on two-dimensional square grids. First, we provide existence and non-existence results for such ground states depending on the values of the nonlinearity powers and on the structure of the set of vertices where the concentrated nonlinearities are located. Second, we prove that suitable piecewise-affine extensions of such states converge strongly in $H^1(\R^2)$ to ground states of corresponding doubly nonlinear models defined on the whole plane as the length of the edges in the grid tends to zero. This convergence is proved both for limit models with standard nonlinearities only and for models combining standard and singular nonlinearities concentrated on a line or on a strip.