Normalized solutions of Schrödinger equations with potentially bounded measures
type: Item
instance of: scholarly article
title: Normalized solutions of Schrödinger equations with potentially bounded measures
zbMATH Open document ID: 1054.31005
author: Wolfhard Hansen
published in: Potential Analysis
publication date: 2004-08-20T00:00:00Z
review text: Let \(\mu\) be a signed Radon measure on a domain \(X\subset \mathbb{R}^n\), \(n\geq 1\), with the Green function \(G_X\). Let \(\mu\) be potentially bounded, i.e., the potential \(G_X^{1_B| \mu| }\) is bounded for every ball \(B\subset X\). In [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28, 413--470 (1999; Zbl 0940.35063)], the present author answered the question of when positive finely continuous solutions of \(\Delta h-h\mu =0\) (\(\equiv \mu\)-harmonic functions) satisfy the Harnack inequality. In particular, it was shown that \(\mu\)-harmonic functions can be very discontinuous. The paper under review is a continuation of that work. Here the author shows that under some general assumptions on \(G\) relative \(\mu\)-harmonic functions \(h/h_0\), where \(h\) and \(h_0\) are \(\mu\)-harmonic and \(h_0\) is strictly positive, are \(\rho\)-Hölder continuous with respect to the natural quasimetric \[ \rho (x,y)=(G(x,y)^{-1}+G(y,x)^{-1})/2. \] The author also studies some perturbations leading to a Brelot space admitting the Green function to be comparable (locally or globally) with \(G_X\).
reviewed by: Alexander I. Kheyfits
Mathematics Subject Classification ID: 31D05
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35J15
zbMATH DE Number: 2093712
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: Hölder continuity
zbMATH Keywords: Green's function
zbMATH Keywords: \(\mu\)-harmonic functions
MaRDI profile type: Publication
full work available at URL: b:pota.0000025378.97262.f3
OpenAlex ID: W2065917400
DOI: 10.1023/B:POTA.0000025378.97262.F3
Recommended article: Criticality for Schrödinger type operators based on recurrent symmetric stable processes
Recommended article: Q4412373
Recommended article: Q3487715
Recommended article: Bĥer's theorem in a space of dimension one
Recommended article: Q3314148
Recommended article: Brelot spaces of Schrödinger equations
Recommended article: Existence and uniqueness of solutions of a semilinear differential equation
Recommended article: Q3701727
Recommended article: Uniqueness Theorems for Subharmonic Functions in Unbounded Domains
Recommended article: Minimal fine limits for a class of potentials
description: scientific article; zbMATH DE number 2093712