Necessary and sufficient conditions for the existence of large solutions to semilinear elliptic equations with gradient terms

Abstract

Our purpose in this paper is to carry out a comprehensive investigation into the question of existence and non-existence of large solutions to semilinear elliptic equations with non-linear gradient terms in bounded domains of the Euclidean space. Based on new Keller-Osserman type integrability criteria, we provide necessary and sufficient conditions for the existence of large solutions. The novelty in our study rests on the fact that the Keller-Osserman type conditions we employ take both the growth rates of the gradient term as well as those of the first order non-linearity terms into account. When the gradient term does not appear, our criteria reduces to the well-known Keller-Osserman condition.