The existence and non-existence of positive solutions to a singular quasilinear elliptic problem in [formula omitted

Abstract

This paper deals with the existence and nonexistence of entire positive solutions of the quasilinear elliptic equation − Δ p u = ρ a ( x ) g ( u ) + λ b ( x ) f ( u ) in R N , 1 < p < N , N ≥ 3 , with u ( x ) → 0 as | x | → ∞ . Here either g or f (or both of them) are singular at 0 in the sense that g ( t ) , f ( t ) → ∞ as t → 0 . By using a perturbation method which eliminates the singularity on a ball with radius R and then let R tends to infinity, with help of the bounded super-solution of the original problem, we obtain the existence of a weak solution of the problem. The main results of this paper improve the corresponding results of [Dragos-Patru Covei, the existence and asymptotic behavior of a positive solution to a quasilinear elliptic problem in R N , Nonlinear Anal. 69 (2008) 2615–2622; J.V.Goncalves, C.A.Santos, Singular elliptic problems: Existence, non-existence and boundary behavior, Nonlinear Anal. 66 (2007) 2078–2090].