Oscillation of Semilinear Elliptic Inequalities by Riccati Transformations

Abstract

A generalized Riccati transformation will be utilized to derive a Riccati-type inequality (3) associated with a semilinear elliptic inequality yL(y; x) ≦ 0 possessing a positive solution y in an exterior domain in Euclidean n-space. On the basis of (3), general sufficient conditions for the elliptic inequality to be oscillatory are developed in § 3. The matrix of coefficients of the second derivative terms in L(y;x) (i.e. (Aij) in (1)) is not restricted in any way beyond the usual ellipticity hypothesis (iv) below, and thereby one of the difficulties mentioned in [9] and inherent in the method there is resolved. Furthermore, the nonlinear term B﹛x, y) in (1) is not required to be one-signed.