Large solutions of semilinear elliptic equations under the Keller-Osserman condition

We consider the equation Delta u = p(x) f (u) where p is a nonnegative nontrivial continuous function and f is continuous and nondecreasing on [0, infinity), satisfies f (0) = 0, f (s) > 0 for s > 0 and the Keller-Osserman condition integral(infinity)(1) (F(s))(-1/2) ds = infinity where F(s) = integral(s)(0)f (t) dt. We establish conditions on the function p that are necessary and sufficient for the existence of positive solutions, bounded and unbounded, of the given equation. Published by Elsevier Inc.