Positive Solutions for the Discrete Nonlinear Schrödinger Equation: A Priori Estimates and Convergence

We consider the discrete nonlinear stationary Schrödinger equation on a bounded n-dimensional box and on the whole space. In the first case we derive the existence of a positive classical solution of the corresponding continuous problem from a uniform a priori bound on positive discrete solutions for a general right hand side. In the second case we derive a uniform a priori bound on positive discrete solutions for the Schrödinger-type nonlinearity.