A Comparison Result and Elliptic EquationsInvolving Subcritical Exponents

It is well known that good bounds for solutions of nonlinear diierential equations are diicult to obtain. In this paper, we establish a theorem comparing non-negative solutions (having identical initial values) of the equations u 00 (t) + q(t)u p (t) + r(t)u(t) = 0 and v 00 (t) + k(t)q(t)v p (t) + r(t)u(t) = 0, respectively. If q(t); r(t) 0, k(t) 1, k(t) is non-decreasing, and the rst equation satisses a certain uniqueness criterion, our result asserts that u(t) v(t). Both the uniqueness assumption on the equation and the monotonicity requirement on k(t) are necessary. A particular case of this theorem plays a central role in a recent paper of Atkinson and Peletier in the study of asymptotic behavior of nonlinear elliptic equations involving a critical exponent. A simple corollary of our result provides information on the same type of equations with subcritical exponents.