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

We consider the equation u = p(x)f (u) where p is a nonnegative nontrivial continuous function and f is continuous and nondecreasing on [0,∞), satisfies f (0) = 0, f (s)>0 for s >0 and the Keller–Osserman condition ∞1 (F (s))−1/2 ds=∞where F(s) = 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.