Non-existence and existence of entire solutions for a quasi-linear problem with singular and super-linear terms

Abstract

We establish results concerning non-existence and existence of entire positive solutions for the nonlinear elliptic problem { − Δ p u = a ( x ) u m + λ b ( x ) u n in R N , u > 0 in R N , u ( x ) ⟶ | x | → ∞ 0 , where − ∞ < m < p − 1 < n with 1 < p < N ; a , b ≥ 0 , a , b ≠ 0 are locally Hölder continuous functions and λ ≥ 0 is a real parameter. The main purpose of this paper, in short, is to complement the principal theorem of B. Xu and Z. Yang (Entire bounded solutions for a class of quasi-linear elliptic equations, Boundary Value Problems 2007, Art. ID 16407, 8 pp) showing existence and non-existence of solutions for the above problem for λ > 0 appropriately.