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

Abstract

We consider the equation Δ 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 d s = ∞ where F ( s ) = ∫ 0 s f ( t ) d t . 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.