We consider the behavior of positive radial solutions (or, briefly, pp.r.s.) of the equation $$ (1.1) ^Delta u + lambda f(u) = 0 inOmega, _u = 0 on partialOmega, $$ where $Omega = {x in R^n$mid$A < $mid$x$mid$ < B}$</TEX> is an annulus in $R^n, n geq 2, lambda > 0 and f geq 0$ is superlinear in u and satisfies f(0) = 0.