Distributional Solutions of the Stationary Nonlinear Schrödinger Equation: Singularities, Regularity and Exponential Decay

  • Rainer Mandel

    Karlsruhe Institute of Technology (KIT), Germany
  • Wolfgang Reichel

    Karlsruhe Institute of Technology (KIT), Germany

Abstract

We consider the nonlinear Schr\"{o}dinger equation

in where the spectrum of is positive. In the case we use variational methods to prove that for all there exist distributional solutions with a point singularity at the origin provided is sufficiently small and are bounded on and satisfy suitable H\"{o}lder-type conditions at the origin. In the case or , however, we show that every distributional solution of the more general equation is a bounded strong solution if is bounded and satisfies certain growth conditions.

Cite this article

Rainer Mandel, Wolfgang Reichel, Distributional Solutions of the Stationary Nonlinear Schrödinger Equation: Singularities, Regularity and Exponential Decay. Z. Anal. Anwend. 32 (2013), no. 1, pp. 55–82

DOI 10.4171/ZAA/1474