JournalszaaVol. 32, No. 1pp. 55–82

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

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

Δu+V(x)u=Γ(x)up1u-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u

in Rn\mathbb R^n where the spectrum of Δ+V(x)-\Delta+V(x) is positive. In the case n3n\geq 3 we use variational methods to prove that for all p(nn2,nn2+ϵ)p\in \big(\frac{n}{n-2},\frac{n}{n-2}+\epsilon \big) there exist distributional solutions with a point singularity at the origin provided ϵ>0\epsilon >0 is sufficiently small and V,ΓV,\Gamma are bounded on RnB1(0)\mathbb R^n\setminus B_1(0) and satisfy suitable H\"{o}lder-type conditions at the origin. In the case n=1,2n=1,2 or n3,1<p<nn2n\geq 3,1<p<\frac{n}{n-2}, however, we show that every distributional solution of the more general equation Δu+V(x)u=g(x,u)-\Delta u + V(x) u = g(x,u) is a bounded strong solution if VV is bounded and gg 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