Positive solutions and spectral properties of second order elliptic operators

Walter Allegretto

Vol. 92 (1981), No. 1, 15–25

DOI: 10.2140/pjm.1981.92.15

Abstract

Let l denote a second order elliptic expression in divergence form and with coefficients defined in an exterior domain. In this paper it is shown that, under suitable conditions, the equation lv = 0 has an a.e. positive generalized solution v defined in a neighborhood of infinity. This is done under weaker conditions on the coefficients of l than was previously required. It is then shown that the existence of such a v implies the finiteness of the negative spectrum of operators naturally associated with l.

Mathematical Subject Classification 2000

Primary: 35J15

Secondary: 35P05

Milestones

Received: 11 April 1979

Published: 1 January 1981