Mathematische Annalen

, Volume 314, Issue 3, pp 555–590

Maximum and anti-maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations

  • Yehuda Pinchover

DOI: 10.1007/s002080050307

Cite this article as:
Pinchover, Y. Math Ann (1999) 314: 555. doi:10.1007/s002080050307

Abstract.

In this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations. We continue the study of some different notions of “small” perturbations and discuss their relations to comparisons of Green's functions, refined maximum and anti-maximum principles, ground state, and the decay of eigenfunctions. In particular, we show that if V is a positive function which is a semismall perturbation of a subcritical Schrödinger operator H defined on a domain \(\Omega\subset \mathbb{R}^d\), and \(\{\phi_k\}_{k\geq 0}\) are the (Dirichlet) eigenfunctions of the equation \(Hu=\lambda Vu\), then for any \(k\geq 0\), the function \(\phi_k/\phi_0\) is bounded and has a continuous extension up to the Martin boundary of the pair \((\Omega, H)\), where \(\phi_0\) is the ground state of H with a principal eigenvalue \(\lambda_0\).

Mathematics Subject Classification (1991):35B05, 35B50, 35J10, 31C35, 35J15.

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Yehuda Pinchover
    • 1
  1. 1.Department of Mathematics, Technion, 32000 Haifa, Israel, (e-mail: pincho@tx.technion.ac.il) IL