We study the following quasilinear elliptic equation ( P β ) − Δ p u + ( β Φ ( x ) − a ( x ) ) u p − 1 + b ( x ) g ( u ) = 0 in R N , where p > 1 , a , b ∈ L ∞ ( R N ) , β , b , g ≥ 0 , b ≢ 0 and Φ ∈ L loc ∞ ( R N ) , inf R N Φ > − ∞ . We provide a sharp criterion in term of generalized principal eigenvalues for existence/nonexistence of positive solution of ( P β ) in suitable classes of functions. Uniqueness result for ( P β ) in those classes is also derived. Under additional conditions on Φ, we further show that: i) either for every β ≥ 0 nonexistence phenomenon occurs, ii) or there exists a threshold value β ⁎ > 0 in the sense that for every β ∈ [ 0 , β ⁎ ) existence and uniqueness phenomenon occurs and for every β ≥ β ⁎ nonexistence phenomenon occurs. In the latter case, we study the limits, as β → 0 and β → β ⁎ , of the sequence of positive solutions of ( P β ) . Our results are new even in the case p = 2 .