Existence of Explosive Solutions to Non-Monotone Semilinear Elliptic Equations

We consider the semilinear elliptic equation u = p(x)f(u) on a domain Rn, n 3, where f is a nonnegative function which vanishes at the origin and satis es g1 f g2 where g1; g2 are nonnegative, nondecreasing functions which also vanish at the origin, and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in such that p is positive on its boundary. For bounded, we show that a nonnegative solution u satisfying u(x) ! 1 as x ! @ exists provided the function (s) Rs 0 f(t) dt satis es R1 1 [ (s)] 1=2 ds lt; 1. For unbounded (including = Rn), we show that a similar result holds where u(x) ! 1 as jxj ! 1 within and u(x) ! 1 as x ! @ if p(x) decays to zero rapidly as jxj ! 1.