Advertisement

Mathematische Annalen

, Volume 315, Issue 4, pp 735–769 | Cite as

On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions

  • Z.-Q. Chen
  • R.J. Williams
  • Z. Zhao
Original article

Abstract.

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains \(D \subset \Bbb R^n\) \((n\ge 3)\) with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form \({1\over 2} \Delta + b\cdot \nabla\) on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an \(L^p\) condition on b, to a Kato condition on \(|b|^2\).

Mathematics Subject Classification (1991):35J65, 60J65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Z.-Q. Chen
    • 1
  • R.J. Williams
    • 2
  • Z. Zhao
    • 3
  1. 1.Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA (e-mail: zchen@math.washington.edu) US
  2. 2.Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA (e-mail: williams@math.ucsd.edu) US
  3. 3.Department of Mathematics, University of Missouri, Columbia, MO 65211, USA (e-mail: zzhao@math.missouri.edu) US

Personalised recommendations